arbres aléatoires, conditionnement et cartes planaires - DMA - Ens

THÈSE DE DOCTORAT DE L’UNIVERSITÉ PARIS 6
Spécialité
Mathématiques
Présentée par
Mathilde Weill
Pour obtenir le grade de
DOCTEUR de l’UNIVERSITÉ PARIS 6
Sujet de la thèse :
ARBRES ALÉATOIRES, CONDITIONNEMENT
ET
CARTES PLANAIRES
soutenue le 8 Décembre 2006 devant le jury composé de
M. Jean Bertoin Examinateur
M. Philippe Chassaing Rapporteur
M. Thomas Duquesne Examinateur
M. Jean-François Le Gall Directeur de thèse
M. Wendelin Werner Examinateur

Remerciements
Mes remerciements vont tout d’abord à Jean-François Le Gall qui a dirigé ma thèse pendant
ces trois dernières années. Il m’a guidée attentivement tout au long de mon travail, prodigué de
nombreux conseils, patiemment expliqué et appris beaucoup de mathématiques. Son exigence
de rigueur et de clarté ont été et resteront pour moi un modèle. Si j’ajoute que son cours sur
le mouvement brownien est l’un de ceux qui m’ont donné le goût des probabilités lors de ma
première année d’études à l’ENS, cela ne donnera encore qu’une faible idée de ce que le présent
travail lui doit.
Philippe Chassaing et Jim Pitman ont accepté de rapporter ma thèse et je leur en suis très
reconnaissante. Philippe Chassaing a de plus accepté de participer au jury, ce dont je le remercie
tout particulièrement.
Jean Bertoin est le second responsable de mon goût pour les probabilités. Son cours “Probabilités
2” est l’un des meilleurs souvenirs de ma première année d’études à l’ENS. Sa présence
dans le jury est un grand plaisir pour moi et je l’en remercie chaleureusement.
Thomas Duquesne et Wendelin Werner ont accepté de faire partie du jury et je les en remercie
vivement. L’intérêt amical qu’ils ont manifesté pour ma thèse tout au long de son élaboration
m’a été précieux.
Je tiens ici à remercier tous les membres du département de mathématiques de l’ENS, qui m’a
offert un cadre idéal pour la rédaction de cette thèse : Patricia qui partage mon bureau depuis
trois ans ; Florent, Nathanaël et Mathieu en souvenir du mois de Juillet 2003 ; mes collègues
caïmans et en particulier Marie, Arnaud et Raphaël ; les voisins du “passage vert” Gilles et les
Philippe; Benoît, David, Frédéric et Guy du “passage bleu”, François et Marc... Je souhaite
aussi remercier Bénédicte, Lara, Laurence et Zaïna pour leur aide, et le spi pour sa disponibilité.
J’adresse enfin toute ma gratitude aux élèves pour avoir rendu mon travail de caïman si agréable.
Je n’oublie pas les thésards et post-doctorants du laboratoire de Paris 6 que j’ai croisés
pendant ces années : Christina, Eulalia, Emmanuel et les autres.
Je termine par une pensée pour mes amis, les matheux : Béné et Greg, et les non-matheux :
Hélhél, Olivia et Babeth ; et bien sûr pour toute ma famille, en particulier mes parents, mon
frère Romain, et Thierry, pour leur soutien constant.
3

Table des matières
Chapitre 1. Introduction 7
1.1. Arbres généalogiques aléatoires 7
1.2. Arbres continus régénératifs 11
1.3. Arbres spatiaux aléatoires 12
1.4. Arbre brownien conditionné 14
1.5. Arbres spatiaux et cartes planaires 18
1.6. Résultats asymptotiques pour de grandes cartes biparties enracinées aléatoires 23
Chapitre 2. Regenerative real trees 27
2.1. Introduction 27
2.2. Preliminaries 29
2.3. Proof of Theorem 2.1.1 35
2.4. Proof of Theorem 2.1.2 44
Chapitre 3. Conditioned Brownian trees 51
3.1. Introduction 51
3.2. Preliminaries 55
3.3. Conditioning and re-rooting of trees 64
3.4. Other conditionings 75
3.5. Finite-dimensional marginal distributions under N 0 83
Chapitre 4. Asymptotics for rooted planar maps and scaling limits of two-type spatial
trees 89
4.1. Introduction 89
4.2. Preliminaries 90
4.3. A conditional limit theorem for two-type spatial trees 98
4.4. Separating vertices in a 2κ-angulation 117
Bibliographie 123
5

CHAPITRE 1
Introduction
Ce travail de thèse est consacré à l’étude d’arbres aléatoires. Dans l’introduction à ce travail,
nous présenterons dans la partie 1.1 les arbres généalogiques aléatoires, c’est-à-dire les arbres
décrivant la généalogie d’une population aléatoire. Puis dans la partie 1.3, nous enrichirons la
structure d’arbres généalogiques pour en faire des arbres spatiaux. Un arbre spatial sera alors
un arbre généalogique pour lequel on a attribué à chacun de ses sommets une position spatiale.
Enfin dans la partie 1.5, nous présenterons les liens entre certains arbres spatiaux et les cartes
planaires biparties.
Les parties 1.2, 1.4 et 1.6 présentent les contributions originales de ce travail de thèse.
1.1. Arbres généalogiques aléatoires
1.1.1. Arbres de Galton-Watson. Un arbre de Galton-Watson est un arbre discret
aléatoire qui décrit la généalogie d’une population gouvernée par un processus de Galton-Watson
(ou processus de branchement discret). Ce modèle d’arbres généalogiques a été introduit par Neveu
[48].
Commençons par présenter le formalisme des arbres discrets. On note U l’ensemble des mots
d’entiers défini par
U = ⋃ n≥0
N n ,
où par convention N = {1,2,...} et N 0 = {∅}. Un élément de U est une suite u = u 1 ...u n
et l’on pose |u| = n de sorte que |u| représente la génération de u. En particulier, |∅| = 0. De
plus, si u = u 1 ...u n ∈ U \ {∅} = U ∗ alors on note ǔ le père de u c’est-à-dire ǔ = u 1 ... u n−1 .
Enfin si u = u 1 ...u n ∈ U et v = v 1 ...v m ∈ U, on définit la concaténation de u et v par
uv = u 1 ...u n v 1 ...v m . En particulier u∅ = ∅u = u.
Un arbre planaire est un sous-ensemble fini A de U vérifiant les propriétés suivantes :
• ∅ ∈ A,
• si u ∈ A \ {∅} alors ǔ ∈ A,
• pour tout u ∈ A, il existe un nombre k u (A) ≥ 0 tel que uj ∈ A si et seulement si
1 ≤ j ≤ k u (A).
On note A l’ensemble des arbres planaires. Si A ∈ A, on note H(A) la hauteur de A c’est-à-dire
H(A) = max{|u| : u ∈ A}.
Tout arbre planaire est codé par un processus appelé processus de contour. Pour définir le
processus de contour d’un arbre planaire A, imaginons une particule se déplaçant tout autour
de A dans le sens des aiguilles d’une montre à vitesse 1. Chaque arête est visitée deux fois par la
particule si bien qu’il lui faut un temps 2(#A − 1) pour parcourir entièrement A. Pour chaque
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t ∈ [0,2(#A − 1)] entier, on définit C(t) comme la hauteur du sommet visité par la particule au
temps t, puis on interpole linéairement C sur l’intervalle [0,2(#A − 1)].
Il existe une deuxième façon de coder un arbre planaire. Si A est un arbre planaire, énumérons
les sommets de A dans l’ordre lexicographique
u(0) = ∅ ≺ u(1) ≺ ... ≺ u(#A − 1).
Pour chaque n ∈ {0,1,... ,#A − 1}, on définit H n comme la hauteur du sommet u(n). Le
processus H = (H n , 0 ≤ n ≤ #A − 1) est appelé processus des hauteurs de A.
Fig. 1. Un arbre, son processus de contour et son processus des hauteurs.
Nous sommes maintenant en mesure de définir les arbres de Galton-Watson. Si A est un arbre
planaire et u ∈ A, on note τ u A = {v ∈ U : uv ∈ A} le sous-arbre de A issu de u. Soit µ une loi
de reproduction critique ou sous-critique, ce qui signifie que ∑ k≥1 kµ(k) ≤ 1. Alors il existe une
unique mesure de probabilité Π µ sur A vérifiant les deux propriétés suivantes :
• Π µ (k ∅ (A) = k) = µ(k) pour tout k ≥ 0,
• si µ(k) > 0, sous la mesure Π µ (· | k ∅ (A) = k), les sous arbres τ 1 A,...,τ k A sont
indépendants et de loi Π µ .
La mesure Π µ est par définition la loi d’un arbre de Galton-Watson de loi de reproduction µ. La
deuxième propriété est une propriété de régénération. Les arbres de Galton-Watson sont ainsi
les seuls arbres discrets régénératifs. Pour p ≥ 1 et A ∈ A, on pose
A p = A ∩ {u ∈ U : |u| ≤ p}.
Soit p ≥ 1, soit T ∈ A tel que H(T) = p et tel que v 1 ≺ ... ≺ v m est la liste des sommets de T à
la génération p énumérés dans l’ordre lexicographique. On a alors plus généralement la propriété
de régénération suivante :
• si Π µ (A p = T) > 0, sous la mesure Π µ (· | A p = T), les sous arbres τ v1 A,... ,τ vm A sont
indépendants et de loi Π µ .
Terminons cette sous-partie en présentant un lien entre les arbres de Galton-Watson et les
marches aléatoires (voir par exemple [41]). Si µ est une loi de reproduction, on définit une mesure
ν sur {−1,0,1,...} par la formule
ν(k) = µ(k + 1), k ≥ −1.
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On peut alors montrer qu’il existe une marche aléatoire (X n ) n≥0 de loi de sauts ν telle que sous
Π µ , presque sûrement, pour tout n ∈ {1,... ,#A − 1},
{
}
H n = # j ∈ {1,... ,n} : X j = inf X k .
j≤k≤n
1.1.2. Arbres continus aléatoires. En vue d’étudier la structure limite de certains arbres
discrets aléatoires, Aldous a introduit la notion d’arbre continu aléatoire.
L’ensemble des arbres continus que nous considérerons est l’ensemble des arbres réels compacts
enracinés. Un arbre réel est un espace métrique (T ,d) tel que pour chaque couple de points
(σ 1 ,σ 2 ) de T , il existe un unique arc noté [[σ 1 ,σ 2 ]] reliant σ 1 à σ 2 , cet arc étant isométrique à un
segment. Dans la suite, nous ne considérerons que des arbres réels compacts et enracinés, c’està-dire
ayant un point ρ distingué, que nous appellerons la racine. Deux arbres réels compacts et
enracinés T et T ′ de racines respectives ρ et ρ ′ sont dits isométriques s’il existe une isométrie
φ de T sur T ′ telle que φ(ρ) = ρ ′ . Nous noterons T l’ensemble des classes d’isométrie d’arbres
réels compacts enracinés.
L’espace T est muni de la distance de Gromov-HausdorffGH définie de la façon suivante :
si T et T ′ sont deux arbres réels compacts de racines respectives ρ et ρ ′ , alors
GH(T , T ′ ) = inf { δ Haus (ϕ(T ),ϕ ′ (T ′ )) ∨ δ(ϕ(ρ),ϕ ′ (ρ ′ )) } ,
où l’infimum est pris sur tous les plongements isométriques ϕ : T → E et ϕ ′ : T ′ → E dans un
même espace métrique (E,δ). On rappelle que si (E,δ) est un espace métrique, on note δ Haus
la distance de Hausdorff définie sur l’ensemble des sous-ensembles compacts de E.
On peut donner une définition équivalente de la distance de Gromov-Hausdorff. Pour ce
faire, il nous faut introduire la notion de correspondance entre deux espaces métriques. Une
correspondance entre deux espaces métriques (E,δ) et (E ′ ,δ ′ ) est un sous-ensemble R de E ×E ′
tel que pour tout x ∈ E (respectivement tout x ′ ∈ E ′ ), il existe x ′ ∈ E ′ (respectivement x ∈ E)
tel que (x,x ′ ) ∈ R. La distorsion de la correspondance R est alors définie par
dis(R) = sup { |δ(x,y) − δ ′ (x ′ ,y ′ )| : (x,x ′ ),(y,y ′ ) ∈ R } .
Si T et T ′ sont deux arbres réels compacts de racines respectives ρ et ρ ′ , on note C(T , T ′ )
l’ensemble des correspondances entre T et T ′ . On a alors
GH(T , T ′ ) = 1 2 inf { dis(R) : R ∈ C(T , T ′ ),(ρ,ρ ′ ) ∈ R } .
Evans, Pitman & Winter [24] ont montré que l’espace (T,GH) est un espace polonais.
On peut construire un arbre réel par un codage similaire au codage d’un arbre planaire par
son processus de contour. Plus précisément, si f : [0,+∞) → [0,+∞) est une fonction continue
à support compact telle que f(0) = 0, on peut construire l’arbre réel compact codé par f de la
façon suivante. Soit d f la pseudo-distance sur R + donnée par la relation
d f (s,t) = f(s) + f(t) − 2 inf
u∈[s,t] f(u),
avec 0 ≤ s ≤ t. On peut alors définir une relation d’équivalence sur R + en posant s ∼ t si
d f (s,t) = 0. L’arbre codé par f est alors l’espace quotient
T f = [0,+∞)/ ∼
muni de la distance d f et enraciné en la classe d’équivalence de 0. Pour tout s ∈ [0,+∞), on
notera ṡ la classe d’équivalence de s. Ainsi ṡ est un sommet de T f à distance f(s) de la racine.
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Remarquons qu’un arbre planaire peut-être interprété comme une union de segments de
longueur 1 de la façon suivante. Soit A ∈ A. On note (ǫ u ,u ∈ A \ {∅}) la base canonique de
R A\{∅} . On définit une famille (l u ,u ∈ A \ {∅}) d’éléments de R A\{∅} par l u = 0 si |u| = 1, et
par récurrence l u = lǔ + ǫǔ si |u| ≥ 2. On pose alors
T A =
⋃
[l u ,l u + ǫ u ].
u∈A\{∅}
On munit T A de la distance d A de sorte que pour tout couple de points (x,y) de T A , la distance
d A (x,y) est la longueur du plus court chemin entre x et y, et l’on enracine T A en 0. Ainsi
(T A ,d A ) est un arbre réel compact enraciné. Par ailleurs, notons C le processus de contour de
A. On vérifie que l’arbre réel T A coïncide avec l’arbre réel T C à isométrie près. Remarquons
que l’arbre planaire A ne peut être reconstruit à partir de l’arbre réel T A . En effet, il n’y a pas
d’ordre entre les enfants d’un même individu dans T A .
Un premier exemple d’arbre continu aléatoire est le CRT d’Aldous [3], [4], [5]. Le CRT
est défini comme l’arbre codé par √ 2e, où e = (e(t),0 ≤ t ≤ 1) est l’excursion brownienne
normalisée. Si T est un arbre réel muni de la distance d, et si r > 0, alors on note rT l’arbre
T muni de la distance rd. On peut alors énoncer un premier résultat de convergence dû à
Aldous. Soit (A n ) n≥1 une suite d’arbres planaires aléatoires telle que chaque A n est uniformément
distribué sur l’ensemble des arbres planaires à n sommets. Alors
n −1/2 T An
(loi)
−→
n→∞ T√ 2e .
Plus généralement, si µ est une loi de reproduction critique et de variance σ 2 finie, alors la loi
de l’arbre réel n −1/2 T A sous Π µ (· | #A = n) converge au sens de la convergence étroite des
mesures sur T vers la loi de l’arbre réel T 2e/σ quand n → ∞.
1.1.3. Arbres de Lévy. Les arbres de Lévy, introduits par Duquesne & Le Gall [22],
sont des analogues continus des arbres de Galton-Watson, au sens où un arbre de Lévy est un
arbre continu aléatoire qui décrit la généalogie d’un processus de branchement à espace d’états
continu.
Soit Y = (Y t ,t ≥ 0) un processus de branchement à espace d’états continu s’éteignant
presque sûrement, de mécanisme de branchement ψ, et soit X un processus de Lévy d’exposant
de Laplace ψ. Duquesne & Le Gall ont montré que l’on peut définir un processus H = (H t ,t ≥ 0)
par la convergence en probabilité suivante, pour tout t ≥ 0,
1
H t = lim
ε→0 ε
∫ t
0½{X s≤inf s≤u≤t X u+ε} ds.
Informellement, H t mesure la taille de l’ensemble {s ∈ [0,t] : X s = inf s≤u≤t X u }. Le processus H
est appelé processus des hauteurs du processus Y par analogie avec le modèle discret. Par ailleurs,
notons I t = inf s∈[0,t] X s pour t ≥ 0, et N la mesure des excursions du processus (X t − I t ,t ≥ 0).
Le processus H peut être défini sous N, et admet sous N une modification continue. L’arbre
de Lévy associé à H est alors l’arbre codé par H sous la mesure N et l’on note Θ ψ la “loi” de
l’arbre T H sous N.
Les arbres stables forment une classe importante d’arbres de Lévy. Ils sont associés aux
mécanismes de branchement ψ(u) = u α pour 1 < α ≤ 2. Duquesne & Le Gall ont calculé la
dimension de Hausdorff d’un arbre stable [22] et ont obtenu des résultats précis concernant la
mesure de Hausdorff d’un arbre stable [23]. Par ailleurs, Miermont [46], [47] et Haas & Miermont
[27] ont utilisé la structure des arbres stables pour étudier les processus de fragmentation stables.
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Enfin, les arbres de Lévy apparaissent comme les seules limites possibles pour des suites
d’arbres de Galton-Watson convenablement changés d’échelle. Nous renvoyons le lecteur aux
theorèmes 2.2.1 et 2.2.3 pour des énoncés précis.
1.2. Arbres continus régénératifs
L’objet de cette partie est de présenter le chapitre 2 de ce travail de thèse, qui est la version
d’un article [53] soumis pour publication. Il s’agit de caractériser les arbres de Lévy par une
propriété de régénération.
Introduisons pour cela quelques notations. Si (T ,d) est un arbre réel compact enraciné en ρ,
on note H(T ) sa hauteur c’est-à-dire
H(T ) = max {d(ρ,σ) : σ ∈ T } .
De plus, pour t,h > 0, on définit Z T (t,t + h) comme le nombre de sous-arbres de T issus du
niveau t et de hauteur strictement supérieure à h. Duquesne & Le Gall [22] ont montré que la
mesure Θ ψ vérifie la propriété de régénération suivante :
(R) pour t,h > 0 et p ∈ N, sous la mesure Θ ψ (· | H(T ) > t) et conditionnellement à
l’événement {Z T (t,t + h) = p}, les p sous-arbres de T issus du niveau t et de hauteur
strictement supérieure à h sont indépendants et de loi Θ ψ (· | H(T ) > h).
La propriété (R) est l’analogue pour des arbres continus de la propriété de régénération des
arbres de Galton-Watson. Nous avons montré que cette propriété caractérise les “lois” des arbres
de Lévy parmi les mesures infinies sur T.
Théorème 1.2.1. Soit Θ une mesure infinie sur (T,GH) telle que Θ(H(T ) = 0) = 0 et
0 < Θ(H(T ) > t) < +∞ pour tout t > 0, et satisfaisant la propriété (R). Alors il existe un
processus de branchement à espace d’états continu s’éteignant presque sûrement, de mécanisme
de branchement ψ, tel que Θ = Θ ψ .
L’idée principale de la preuve du théorème 1.2.1 est d’approcher les arbres régénératifs par
des arbres de Galton-Watson. Cette idée repose sur un procédé de discrétisation des arbres réels
que nous allons décrire brièvement.
On se donne ε > 0, n ≥ 1 et un arbre réel T ∈ T de hauteur H(T ) vérifiant (n + 1)ε <
H(T ) ≤ (n + 2)ε. Pour chaque k ∈ {0,1,... ,n}, on note σ1 k,... ,σk m k
les sommets de T à la
génération kε (c’est-à-dire à distance kε de la racine) dont sont issus les sous-arbres de T au
dessus du niveau kε et de hauteur strictement supérieure à ε. Chaque sommet σi
k+1 appartient à
un sous-arbre issu d’un élément σj k i
de l’ensemble {σl k : 1 ≤ l ≤ m k }. On dit alors que l’individu
σi k+1 est issu de σj k i
. Une difficulté provient du fait que l’ordre entre les individus σi k+1
1
,... ,σi k+1
l j
issus d’un même parent σj k n’est pas défini. En ordonnant uniformément ces individus, on peut
construire un arbre planaire A ε (T ) rendant compte de la généalogie dans l’arbre réel T de la
collection de points {σi k : 1 ≤ i ≤ m k, 1 ≤ k ≤ n}.
La propriété de régénération (R) nous assure alors que si T est distribué selon la mesure
de probabilité Θ(· | (n + 1)ε < H(T ) ≤ (n + 2)ε), l’arbre planaire A ε (T ) est un arbre de
Galton-Watson conditionné à avoir une hauteur égale à n.
Dans ce travail, nous nous sommes également intéressés aux mesures de probabilité sur T
vérifiant la propriété de régénération (R).
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Théorème 1.2.2. Soit Θ une mesure de probabilité sur (T,GH) telle que Θ(H(T ) = 0) = 0
et 0 < Θ(H(T ) > t) pour tout t > 0, et satisfaisant la propriété (R). Alors il existe a > 0 et µ,
une mesure de probabilité critique ou sous-critique sur Z + \ {1} tels que Θ est la loi de l’arbre
généalogique d’un processus de branchement en temps continu, à espace d’états discret, de loi de
reproduction µ et de taux de branchement a.
1.3. Arbres spatiaux aléatoires
1.3.1. Arbres de Galton-Watson spatiaux. Un arbre spatial discret est un couple
(A,U) où A ∈ A et U = (U v ,v ∈ A) est une application de A dans R. On note Ω l’ensemble des
arbres spatiaux discrets.
Rappelons que l’on peut coder un arbre planaire par son processus de contour C. De même,
on peut définir le processus de contour spatial d’un arbre spatial discret (A,U) en posant pour
tout t ∈ [0,2(#A − 1)] entier, V (t) = U v où v est le sommet visité par C au temps t, puis on
interpole linéairement V sur l’intervalle [0,2(#A − 1)]. Le couple de processus (C,V ) code alors
l’arbre spatial (A,U).
Soit µ une loi de reproduction et soit γ une mesure de probabilité sur R. Construisons à
présent sur l’ensemble Ω la loi d’un arbre de Galton-Watson spatial de loi de reproduction µ et
dont les déplacements spatiaux sont régis par γ. Soit A ∈ A et soit x ∈ R. On définit la mesure
R x (A,dU) sur R A de la façon suivante. Considérons une suite (Y u ,u ∈ U) de variables aléatoires
indépendantes et de loi γ. On pose U ∅ = x et pour tout v ∈ A \ {∅},
U v = ∑
Y v ′,
v ′ ∈]∅,v]
où ]∅,v] est l’ensemble des ancêtres de v privé de la racine ∅ (noter que v ∈]∅,v]). La mesure
R x (A,dU) est alors la loi de (U v ,v ∈ A). Posons
P x (dAdU) = Π µ (dA)R x (A,dU).
On dit que la mesure P x est la loi d’un arbre de Galton-Watson spatial issu de x, de loi de
reproduction µ et de loi de déplacement spatial γ.
1.3.2. Arbre brownien et serpent brownien. L’arbre brownien est un arbre réel spatial
aléatoire. Un arbre réel spatial est un couple (T ,Y ) où T est un arbre réel compact et enraciné
et Y = (Y σ ,σ ∈ T ) est une application continue de T dans R. Deux arbres réels spatiaux (T ,Y )
et (T ′ ,Y ′ ) sont dits isométriques si T et T ′ sont isométriques et si pour tout σ ∈ T ,
Y ′ φ(σ) = Y σ,
où φ est une isométrie de T sur T ′ telle que φ(ρ) = ρ ′ . On notera T sp l’ensemble des classes
d’isométrie d’arbres réels spatiaux.
Rappelons que C(T , T ′ ) est l’ensemble des correspondances entre T et T ′ et que si R ∈
C(T , T ′ ), alors on note dis(R) la distorsion de R. On définit sur T sp une distancesp par la
formule suivante :
(
sp (T ,Y ),(T ′ ,Y ′ ) ) {
}
= 1 2 inf dis(R) + sup |Y σ − Y σ ′ ′| : R ∈ C(T , T ′ ),(ρ,ρ ′ ) ∈ R .
(σ,σ ′ )∈R
On vérifie alors que l’espace T sp muni de la distancesp est un espace polonais.
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Nous pouvons maintenant définir l’arbre brownien. Si T ∈ T et x ∈ R, on note Q x (T ,dY ) la
loi du processus gaussien (Y σ ,σ ∈ T ) caractérisé par
• E(Y σ ) = x,
• Cov(Y σ ,Y σ ′) = d(ρ,σ ∧ σ ′ ),
où σ ∧ σ ′ est le plus proche ancêtre commun à σ et σ ′ , c’est-à-dire le sommet de T vérifiant
la relation [[ρ,σ ∧ σ ′ ]] = [[ρ,σ]] ∩ [[ρ,σ ′ ]] (sous une hypothèse faible sur T toujours réalisée dans
la suite, (Y σ ,σ ∈ T ) a une modification continue). Notons n la loi de l’excursion brownienne
normalisée. On définit une mesure de probabilité N x (dT dY ) sur T sp en posant
∫
∫ ∫
N x (dT dY )F(T ,Y ) = n(de) Q x (T e ,dY )F(T e ,Y ).
La mesure N x est la loi de l’arbre brownien issu de x 1 .
Le serpent brownien, introduit par Le Gall, est un objet intimement lié à l’arbre brownien.
Nous renvoyons le lecteur à [37] pour une présentation détaillée du serpent brownien et de
certaines de ses applications. Le serpent brownien est un processus de Markov à valeurs dans
l’espace des trajectoires arrêtées
W = ⋃
C([0,t], R).
t∈R +
Pour tout w ∈ W, on pose ζ w = t si w ∈ C([0,t], R). Autrement dit, ζ w représente le temps
de vie de la trajectoire w. De plus, on note ŵ = w(ζ w ) le point terminal de w. On définit une
distance d W sur W de la façon suivante :
d W (w,w ′ ) = sup
t≥0
∣
∣w(t ∧ ζ w ) − w ′ (t ∧ ζ w ′) ∣ + |ζ w − ζ w ′|.
On peut alors montrer que l’espace W muni de la distance d W est un espace polonais.
Construisons à présent le serpent brownien “conditionnellement à son temps de vie”. Pour
cela, on se donne une fonction continue f : [0,1] −→ [0,+∞) telle que f(0) = 0, et l’on pose
pour tous 0 ≤ s ≤ s ′ ≤ 1,
m(s,s ′ ) = inf
u∈[s,s ′ ] f(u).
Soit x ∈ R. Il existe alors un processus de Markov inhomogène (W s ,0 ≤ s ≤ 1) à valeurs dans W
tel que W 0 = x presque sûrement, et dont le noyau de transition est caractérisé par la description
suivante : pour tous 0 ≤ s ≤ s ′ ≤ 1,
• W s ′(t) = W s (t) pour tout t ≤ m(s,s ′ ), presque sûrement,
• (W s ′(m(s,s ′ ) + t) − W s (m(s,s ′ )),0 ≤ t ≤ f(s ′ ) − m(s,s ′ )) est indépendant de W s et
suit la loi d’un mouvement brownien partant de 0.
On note θ f x la loi du processus (W s ,s ≥ 0) et l’on pose,
N x (dζ dW) = n(dζ)θ ζ x(dW).
Le serpent brownien est alors le processus canonique ((ζ s ,0 ≤ s ≤ 1),(W s ,0 ≤ s ≤ 1)) de
l’espace C([0,1], R + ) × C([0,1], W) muni de la mesure de probabilité N x .
Remarquons que l’on peut définir sous N x un processus Y = (Y σ ,σ ∈ T ζ ) en posant pour
tout s ∈ [0,1] tel que σ = ṡ,
Y σ = Ŵs.
1 Les notations n et Nx introduites ici n’ont pas les mêmes significations que dans le chapitre 3.
13

Rappelons que si s ∈ [0,1], alors ṡ est la classe d’équivalence de s pour la relation d’équivalence
définie par ζ. En outre, on voit que la loi de (T ζ ,Y ) est la mesure N x .
Une mesure aléatoire est naturellement associée à l’arbre brownien. Il s’agit de la mesure Z
sur R définie par
∫ 1
∫ 1 )
〈Z,ϕ〉 = ϕ(Yṡ)ds = ϕ
(Ŵs ds.
0
Cette mesure aléatoire est appelée Integrated Super-Brownian Excursion (ISE). La mesure ISE
en grande dimension intervient dans divers résultats asymptotiques de modèles de mécanique
statistique (voir par exemple [18], [28], [29]).
Enfin, l’arbre brownien apparaît comme limite de suites d’arbres spatiaux discrets convenablement
renormalisés. Plus précisément, Janson & Marckert [31] ont montré le résultat suivant.
Soit µ une loi de reproduction critique telle qu’il existe η > 0 satisfaisant
∑
e ηk µ(k) < +∞.
k≥0
Soit γ une mesure de probabilité sur R, de moyenne nulle et vérifiant la condition de moments
suivante : quand y → +∞,
γ ({|u| > y}) = o ( y −4) .
Les lois µ et γ ont donc des variances finies et l’on pose Var(µ) = ϑ 2 µ et Var(γ) = ϑ 2 γ. Rappelons
que si A est un arbre planaire, on note C le processus de contour de A et V le processus de
contour spatial de A. Alors, pour tout x ∈ R, la loi sous la mesure P x (· | #A = n) de
⎛
⎝(
ϑµ
2
)
C(2nt)
√ n
0≤t≤1
,
(
(
1 ϑµ
ϑ γ 2
0
) )
1/2
V (2nt)
n 1/4
0≤t≤1
converge quand n → ∞ vers la loi sous la mesure de probabilité N 0 du couple
(
)
(ζ s ,0 ≤ s ≤ 1),(Ŵs,0 ≤ s ≤ 1) .
1.4. Arbre brownien conditionné
Nous présentons dans cette partie le chapitre 3 de ce travail de thèse, qui est une version d’un
article [42] écrit en collaboration avec Jean-François Le Gall et paru aux annales de l’institut
Henri Poincaré. Il s’agit de définir l’arbre brownien issu de 0 conditionné à rester positif.
Pour (T ,Y ) ∈ T sp , on note X T ,Y = {Y σ : σ ∈ T }. On cherche donc à conditionner l’arbre
brownien issu de 0 par l’événement {X T ,Y ⊂ [0,+∞)}. Ce conditionnement est un conditionnement
dégénéré car
N 0 (X T ,Y ⊂ [0,+∞)) = 0.
En revanche, pour tout ε > 0, on a
N 0 (X T ,Y ⊂ (−ε,+∞)) > 0.
L’idée est donc de construire la loi de l’arbre brownien conditionné à rester positif comme la
limite en un certain sens des mesures N 0 (dT dY | X T ,Y ⊂ (−ε,+∞)).
Théorème 1.4.1. On a
N 0 (X T ,Y ⊂ (−ε, ∞))
lim
ε↓0 ε 4 = 2
21 .
⎞
⎠
14

De plus, il existe une mesure N 0 sur l’espace T sp telle que
lim
ε↓0
N 0 (dT dY | X T ,Y ⊂ (−ε, ∞)) = N 0 (dT dY ),
au sens de la convergence étroite des mesures sur T sp .
Le deuxième théorème principal de ce travail donne une représentation explicite de l’arbre réel
spatial de loi N 0 au moyen d’une transformation de l’arbre brownien analogue à la transformation
de Vervaat du pont brownien.
Rappelons brièvement en quoi consiste la transformation de Vervaat du pont brownien. Soit
(B t ,t ∈ [0,1]) un pont brownien sur [0,1]. Il est connu que presque sûrement, il existe un unique
instant t ∗ ∈ [0,1] tel que
B t∗ = min
t∈[0,1] B t.
On définit un processus B ∗ = (B ∗ t ,t ∈ [0,1]) par la transformation
B ∗ t = B {t∗+t} − B t∗ ,
où {t ∗ + t} est la partie fractionnaire de t ∗ + t. Vervaat [52] a montré que la loi du processus B ∗
est alors la loi n de l’excursion brownienne normalisée.
La transformation que nous effectuons sur les arbres réels spatiaux est une opération de
“réenracinement au minimum”. Présentons tout d’abord en quoi consiste le réenracinement d’un
arbre réel spatial en l’un de ses sommets. Si (T ,Y ) ∈ T sp et σ ∈ T , on définit l’arbre réenraciné
(T [σ] ,Y [σ] ) de la façon suivante. L’arbre T [σ] est l’arbre T , mais sa racine est le sommet σ et
non plus ρ. Puis on translate les positions spatiales en posant pour tout σ ′ ∈ T ,
Y [σ]
σ ′ = Y σ ′ − Y σ .
Par ailleurs, on montre (voir la proposition 3.2.5) que sous la mesure N 0 , il existe presque
sûrement un unique sommet σ ∗ ∈ T tel que
Y σ∗ = min {Y σ : σ ∈ T }.
On est alors en mesure d’énoncer le deuxième théorème principal de ce travail.
Théorème 1.4.2. La mesure N 0 est la loi sous N 0 de l’arbre réenraciné (T [σ∗] ,Y [σ∗] ).
Le point de départ de la preuve de ce théorème est une propriété d’invariance de l’opération
de réenracinement sous N 0 . Marckert & Mokkadem [45] (voir aussi le théorème 3.2.3) ont montré
que pour toute fonction mesurable positive sur T sp et pour tout s ∈ [0,1], on a
(
N 0
(F T [ṡ] ,Y [ṡ])) = N 0 (F(T ,Y )) .
Les théorèmes 1.4.1 et 1.4.2 peuvent être exprimés en termes du serpent brownien. Tout
d’abord, notons que la proposition 3.2.5 nous assure que N 0 presque sûrement, il existe un
unique s ∗ ∈ [0,1] tel que
}
Ŵ s∗ = min{Ŵs : s ∈ [0,1] = min {W s (t) : t ∈ [0,ζ s ], s ∈ [0,1]} .
De plus, pour tout s ∈ [0,1], on pose
ζ [s]
r = ζ s + ζ {s+r} − 2 inf
[s,{s+r}] ζ,
Ŵ [s]
r
= Ŵ{s+r} − Ŵs.
15

On définit ensuite W [s] à partir de Ŵ [s] de la façon suivante : pour tout r ∈ [0,1] et tout
t ∈ [0,ζ r [s] ],
[s]
(t) = Ŵ n
o.
W [s]
r
sup
u≤r:ζ u [s] =t
Notons X = {Ŵs : s ∈ [0,1]}. On a alors l’existence d’une mesure N 0 sur C(R + , R + )×C(R + , W)
telle que
N 0 (dζ dW | X ⊂ (−ε,+∞)) −→
ε↓0
N 0 (dζ dW),
au sens de la convergence étroite des mesures sur C(R + , R + ) × C(R + , W). De plus N 0 est la loi
sous N 0 du serpent réenraciné (ζ [s∗] ,W [s∗] ).
De même que précédemment, on définit sous N 0 un processus Y = (Y σ ,σ ∈ T ζ ) en posant
pour tout s ∈ [0,1] tel que σ = ṡ,
Y σ = Ŵs.
Alors la loi de (T ζ ,Y ) sous N 0 est la mesure N 0 .
Une des motivations de ce travail était d’exhiber un arbre aléatoire qui serait une limite
possible de suites d’arbres discrets conditionnés à rester positifs, et convenablement renormalisés.
Le Gall [39] a montré le résultat suivant. Notons, pour (A,U) ∈ Ω,
U = min{U v : v ∈ A \ {∅}}.
Supposons que les lois µ et γ vérifient les mêmes hypothèses que celles du résultat de Janson
& Marckert et que la loi γ est symétrique. Alors pour tout x > 0, la loi sous la mesure de
probabilité P x (· | #A = n + 1,U > 0) de
⎛
⎝(
ϑµ
2
)
C(2nt)
√ n
0≤t≤1
,
(
(
1 ϑµ
ϑ γ 2
) )
1/2
V (2nt)
n 1/4
0≤t≤1
converge quand n → ∞ vers la loi sous la mesure de probabilité N 0 du couple
(
)
(ζ s ,0 ≤ s ≤ 1),(Ŵs,0 ≤ s ≤ 1) .
Dans ce travail nous nous sommes également intéressés au conditionnement de l’arbre brownien
ayant une hauteur fixée. Pour h > 0, on note n h la loi de l’excursion brownienne de hauteur
h. On définit une mesure de probabilité N h x (dT dY ) en posant pour x ∈ R,
∫
∫ ∫
N h x(dT dY )F(T ,Y ) = n h (de) Q x (T e ,dY )F(T e ,Y ).
Rappelons que si f : [0,+∞) −→ [0,+∞) est une fonction continue à support compact telle que
f(0) = 0, on dit que f est de hauteur h si
max f(s) = h.
s∈[0,+∞)
Remarquons que si f est de hauteur h, alors H(T f ) = h. On dit que N h x est la loi de l’arbre
brownien de hauteur h issu de x.
Théorème 1.4.3. Pour tout h > 0, il existe une mesure de probabilité N h 0 sur T sp telle que
lim
ε→0 Nh 0(dT dY | X T ,Y ⊂ (−ε,+∞)) = N h 0(dT dY ),
au sens de la convergence étroite des mesures sur T sp .
⎞
⎠
16

Décrivons l’arbre réel spatial de loi N h 0 . Tout d’abord, on peut montrer que Nh 0 presque
sûrement, il existe un unique sommet σ ∈ T tel que
d(ρ,σ) = h.
Pour tout t ∈ [0,h], soit σ t l’unique point de l’arc [[ρ,σ]] tel que d(ρ,σ t ) = t. On définit alors un
processus (R h t ,0 ≤ t ≤ h) par la relation,
R h t = Y σ t
, t ∈ [0,h].
On montre que la loi du processus (Rt h ,0 ≤ t ≤ h) est absolument continue par rapport à la loi
de (β t ,0 ≤ t ≤ h) où β est un processus de Bessel de dimension 9. Nous renvoyons à la partie
3.4 pour l’expression explicite de la densité de la loi de (Rt h ,0 ≤ t ≤ h) par rapport à la loi du
processus de Bessel (β t ,0 ≤ t ≤ h). De plus, on montre que le processus (Rt∧h h ,t ≥ 0) converge
en loi vers (β t ,t ≥ 0) quand h → ∞.
Par ailleurs, notons (T i ,i ∈ I) la famille des sous-arbres de T issus de l’arc [[ρ,σ]]. Soit σ i le
point de [[ρ,σ]] dont est issu le sous-arbre T i . On pose
On définit alors une mesure ponctuelle M par
d i = d(ρ,σ i ).
M = ∑ i∈I
δ (di ,T i ).
Puis, si l > 0, on note n

1.5. Arbres spatiaux et cartes planaires
1.5.1. Lois de Boltzmann sur les cartes planaires. Une carte planaire M est un plongement
d’un graphe connexe G dans la sphère bidimensionnelle S 2 , c’est à dire un “dessin” de
G dans S 2 tel que les arêtes ne se rencontrent qu’au niveau des sommets. Une face de M est
une composante connexe de S 2 \ M, et son degré est le nombre d’arêtes de M incluses dans sa
fermeture. Une carte planaire est dite bipartie si chacune de ses faces est de degré pair. Une
carte planaire est une 2κ-angulation si chacune de ses faces est de degré 2κ.
Si M est une carte planaire, on notera respectivement V M , E M et F M l’ensemble des sommets,
des arêtes et des faces de M. L’ensemble V M est muni de la distance de graphe, c’est-à-dire que la
distance entre deux sommets de M est la longueur du plus court chemin entre ces deux sommets.
Une carte planaire pointée est un couple (M,τ) où M est une carte planaire et τ est un
sommet distingué de M. Une carte planaire enracinée est un couple (M,⃗e ) où M est une carte
planaire et ⃗e est une arête orientée de M. On appelle sommet racine le point dont est issue l’arête
orientée ⃗e. Une carte planaire enracinée et pointée est un triplet (M,τ,e) où (M,τ) est une carte
planaire pointée et e est une arête non orientée de M. Remarquons que l’on peut interpréter une
carte enracinée (M,⃗e ) comme une carte enracinée et pointée en conservant l’arête non-orientée
e et en choisissant le sommet racine comme point distingué.
On identifie deux cartes planaires pointées (respectivement enracinées, enracincées et pointées)
M et M ′ s’il existe un homéomorphisme de S 2 préservant l’orientation, qui envoie M sur M ′ et
qui préserve le point distingué (respectivement l’arête orientée, le point distingué et l’arête nonorientée).
On note respectivement M p , M r et M r,p , l’ensemble des cartes planaires pointées,
enracinées et enracinées et pointées après l’identification précédente.
On définit à partir d’une suite q = (q i ) i≥1 de poids positifs telle que q i > 0 pour au moins un
indice i ≥ 2, une mesure de Boltzmann sur l’ensemble M r,p de la façon suivante. Pour M ∈ M r,p ,
on pose
W q (M) = ∏
q deg(f)/2 ,
f∈F M
où deg(f) est le degré de la face f. Si la suite q vérifie
Z q =
∑
W q (M) < ∞,
M∈M r,p
on dit que q est admissible et l’on définit la distribution de Boltzmann B r,p
q sur l’ensemble M r,p
par
B r,p
q (M) = W q(M)
.
Z q
De plus, on définit pour M ∈ M r
Z q (r) = ∑ ∏
q deg(f)/2 .
f∈F M
La condition Z q < ∞ implique que Z q
(r)
sur l’ensemble M r par
M∈M r
< ∞. Alors on définit la distribution de Boltzmann B r q
B r q(M) = W q(M)
Z q
(r) .
18

On pose N(k) = ( 2k−1
k−1
)
pour k ≥ 1. Pour tout suite q de poids comme précédemment on
définit
f q (x) = ∑ k≥0
N(k + 1)q k+1 x k , x ≥ 0,
et l’on note R q le rayon de convergence de la série entière f q . D’après la proposition 1 dans [43],
la suite q est admissible si et seulement si l’équation
(1.5.1) f q (x) = 1 − 1 x , x > 0,
admet au moins une solution. Dans ce cas Z q est la solution de (1.5.1) vérifiant
On dit que la suite q est régulière si
Z 2 q f ′ q (Z q) ≤ 1.
Z 2 q f ′ q (Z q) = 1.
Si de plus Z q < R q on dit que q est critique régulière.
Présentons finalement un cas particulier de mesure de Boltzmann ne chargeant que l’ensemble
des 2κ-angulations. Soit κ ≥ 2. On pose
α κ =
(κ − 1)κ−1
κ κ N(κ) .
Soit q κ la suite de poids définie par q κ = α κ et q i = 0 pour tout i ∈ N \ {κ}. D’après la partie
1.5 dans [43], la suite q κ est critique régulière et
Z qκ =
κ
κ − 1 .
Pour tout n ≥ 1, notons respectivement U n κ et U n κ la mesure uniforme sur l’ensemble des 2κangulations
enracinées et pointées et la mesure uniforme sur l’ensemble des 2κ-angulations enracinées.
On voit alors que
B r,p
q κ
(· | #F M = n) = U n κ,
B r q κ
(· | #F M = n) = U n κ .
1.5.2. Loi de Boltzmann sur l’ensemble des mobiles. Dans cette partie, on interprétera
les arbres spatiaux comme des arbres spatiaux à deux types en déclarant que les sommets
des générations paires sont de type 0 et les sommets des générations impaires sont de type 1.
Pour (A,U) ∈ A, on pose
A 0
A 1
= {u ∈ A : |u| est pair},
= {u ∈ A : |u| est impair}.
Un mobile (enraciné) est un arbre spatial à deux types (A,U) tel que les propriétés suivantes
sont vérifiées :
(a) U v = Uˇv pour tout v ∈ A 1 .
(b) Soit v ∈ A 1 tel que k v (A) = k ≥ 1. On note v (0) = ˇv le père de v et v (j) = vj pour tout
j ∈ {1,... ,k}. Alors pour tout j ∈ {0,... ,k},
où par convention v (k+1) = v (0) .
U v(j+1) ≥ U v(j) − 1,
19

Si de plus U v ≥ 1 pour tout v ∈ A, alors (A,U) est un mobile bien étiqueté.
Construisons à présent une loi de Galton-Watson sur l’ensemble des mobiles de la façon
expliquée dans [43]. Soit q une suite de poids critique régulière. Notons µ q 0 la loi géométrique
de paramètre f q (Z q ) c’est-à-dire la mesure sur {0,1,2,...} définie pour k ≥ 0 par
et µ q 1
µ q 0 (k) = f q(Z q ) k
Z q
,
la mesure sur {0,1,2,...} définie pour k ≥ 0 par
µ q 1 (k) = Zk qN(k + 1)q k+1
.
f q (Z q )
On pose µ q = (µ q 0 ,µq 1 ). On définit une mesure de probabilité P µ q sur l’ensemble des arbres
planaires par la formule suivante : pour t ∈ A,
P µ q(t) = ∏ u∈t 0 µ q 0 (k u(t)) ∏ u∈t 1 µ q 1 (k u(t)).
Pour tout k ≥ 1 on note ν0 k la mesure de Dirac en 0 ∈ Rk et ν1 k la mesure uniforme sur
l’ensemble A k défini par
{
}
A k = (x 1 ,...,x k ) ∈ Z k : x 1 ≥ −1,x 2 − x 1 ≥ −1,... ,x k − x k−1 ≥ −1, −x k ≥ −1 .
La mesure ν1 k est alors la loi du vecteur (X 1,X 1 + X 2 ,...,X 1 + ... + X k ) où (X 1 ,...,X k+1 ) est
uniformément distribué sur l’ensemble B k defini par
{
}
B k = (x 1 ,...,x k+1 ) ∈ {−1,0,1,2,...} k+1 : x 1 + ... + x k+1 = 0 .
Soit A ∈ A. On définit une mesure de probabilité R ν,1 (A,dU) sur R A de la façon suivante.
Soit (Y u ,u ∈ A) une suite de vecteurs aléatoires indépendants telle que pour u ∈ A vérifiant
k u (A) = k, le vecteur Y u = (Y u1 ,... ,Y uk ) est de loi ν0 k si u ∈ A0 et de loi ν1 k si u ∈ A1 . On
définit U ∅ = 1 et pour tout v ∈ A \ {∅},
U v = 1 + ∑
Y u ,
u∈]∅,v]
où ]∅,v] représente l’ensemble des ancêtres de v privé de la racine ∅. La mesure R ν,1 (A,dU)
est alors la loi de (U v ,v ∈ A).
Soit P µ,ν,1 la mesure de probabilité définie par
P µ,ν,1 (dAdU) = P µ (dA)R ν,1 (A,dU).
On vérifie facilement que la mesure P µ,ν,1 ne charge que l’ensemble des mobiles (A,U) tels que
U ∅ = 1.
1.5.3. La bijection de Bouttier, di Francesco & Guitter. Bouttier, di Francesco &
Guitter [11] ont établi une bijection entre l’ensemble des mobiles bien étiquetés et l’ensemble
des cartes biparties enracinées. Cette bijection est une généralisation de la bijection construite
par Schaeffer [51] entre les arbres bien étiquetés et les quadrangulations enracinées (voir aussi
[15] pour une étude antérieure de cette bijection).
Présentons dans un premier temps une version de la bijection de Bouttier, di Francesco &
Guitter entre l’ensemble des mobiles et l’ensemble des cartes biparties enracinées et pointées.
Soit (A,U) un mobile tel que U ∅ = 1. On pose ξ = #A−1 et l’on définit une suite w 0 ,w 1 ,... ,w ξ
20

de sommets de A 0 de telle sorte que pour tout k ∈ {0,1,... ,ξ}, le sommet w k est le sommet
visité par le contour de A au temps 2k. De plus on définit un mobile bien étiqueté (A,U + ) par
la relation pour v ∈ A
U + v = U v − min{U w : w ∈ A} + 1.
Notons que
min { U + w : w ∈ A} = 1.
Supposons maintenant que l’arbre planaire A est dessiné dans le plan et ajoutons un sommet
∂. On associe au mobile (A,U) une carte planaire dont l’ensemble des sommets est
A 0 ∪ {∂},
et dont les arêtes sont obtenues par le procédé suivant :
• si U + w k
= 1, on trace une arête entre w k et ∂,
• si U + w k
≥ 2, on trace une arête entre w k et le premier sommet parmi w k+1 ,...,w ξ−1 ,
w 0 ,...,w k−1 dont l’étiquette est U + w k
− 1.
La condition (b) de la définition d’un mobile garantit que si U + w k
≥ 2, alors il existe un sommet
parmi w k+1 ,...,w ξ−1 ,w 0 ,...,w k−1 dont l’étiquette est U + w k
− 1. La carte planaire ainsi obtenue
est une carte bipartie que l’on pointe en ∂ et que l’on enracine en l’arête obtenue pour k = 0
dans la construction précédente.
Bouttier, di Francesco & Guitter [11] ont montré que la construction précédente fournit une
bijection Ψ r,p entre l’ensemble des mobiles (A,U) tels que U ∅ = 1 et l’ensemble des cartes
biparties enracinées et pointées.
On remarque que si (A,U) est un mobile bien étiqueté tel que U ∅ = 1 alors U + v = U v pour
tout v ∈ A. En particulier U + ∅ = 1. On en déduit que l’arête distinguée de la carte Ψ r,p((A,U))
contient le point ∂. On peut ainsi indentifier Ψ r,p ((A,U)) à une carte enracinée. Cela implique
que la bijection Ψ r,p induit une bijection Ψ r entre l’ensemble des mobiles bien étiquetés (A,U)
tels que U ∅ = 1 et l’ensemble M r (voir la figure 2).
La bijection Ψ r vérifie les deux propriétés suivantes : soit (A,U) un mobile tel que U ∅ = 1
et soit M = Ψ((A,U)),
• pour tout k ≥ 1, l’ensemble {f ∈ F M : deg(f) = 2k} est en bijection avec l’ensemble
{v ∈ A 1 : k v (A) = k},
• pour tout l ≥ 1, l’ensemble {a ∈ V M : d(∂,a) = l} est en bijection avec l’ensemble
{v ∈ A 0 : U v = l}.
Enfin signalons la propriété suivante (voir la proposition 10 dans [43]). La distribution de
Boltzmann conditionnée B r,p
q (· | #F M = n) est l’image de la mesure P µ q ,ν,1(· | #A 1 = n) par
l’application Ψ r,p . De plus, pour tout mobile (A,U) on définit
U = min { U v : v ∈ A 0 \ {∅} } .
La distribution de Boltzmann conditionnée B r q (· | #F M = n) est alors l’image de la mesure
P µ q ,ν,1(· | #A 1 = n, U ≥ 1) par l’application Ψ r .
21

Fig. 2. Construction d’une carte bipartie enracinée à partir d’un mobile bien étiqueté
1.5.4. Un théorème limite pour les mobiles. En vue d’étudier certaines propriétés
asymptotiques des grandes cartes biparties enracinées et pointées, Marckert & Miermont [43]
ont établi un théorème limite pour les mobiles.
Avant d’énoncer ce résultat, rappelons que si (A,U) est un arbre spatial, on note C son
processus de contour et V son processus de contour spatial. De plus, pour toute suite de poids
critique régulière q, on pose
ρ q = 2 + Zqf 3 q(Z ′′ q ).
Marckert & Miermont [43] ont montré le théorème suivant : si q est une suite critique régulière,
alors la loi sous la mesure P µ q ,ν,1(· | #A 1 = n) de
( (αq ( )
C(2(#A − 1)t) βq V (2(#A − 1)t)
n
)0≤t≤1
1/2 ,
n
)0≤t≤1
1/4
converge (au sens de la convergence étroite des mesures sur C([0,1], R 2 ) quand n → ∞ vers la
loi de (ζ,Ŵ) sous la mesure N 0, où les constantes α q et β q sont définies par les formules
√
ρq (Z q − 1)
α q =
,
4
( ) 9(Zq − 1) 1/4
β q =
.
4ρ q
Ce résultat est un cas particulier du théorème 11 dans [43] qui traite d’arbres de Galton-Watson
spatiaux à deux types plus généraux. Signalons que dans le cas particulier q = q κ les constantes
apparaissant dans le résultat précédent sont données par
α qκ = 1 ( ) κ 1/2
,
4 κ − 1
β qκ =
(
9
4κ(κ − 1)
) 1/4
.
22

1.6. Résultats asymptotiques pour de grandes cartes biparties enracinées
aléatoires
Cette partie récapitule les résultats du chapitre 4 de ce travail de thèse, dans lequel nous nous
intéressons à certaines propriétés asymptotiques de grandes cartes planaires biparties enracinées
aléatoires.
Introduisons au préalable quelques notations. Soit M ∈ M r . On note o le sommet racine de
M c’est-à-dire le sommet dont est issue l’arête racine. Soit
R M = max {d(o,a) : a ∈ V M } ,
le rayon de la carte M. Le profil de M est la mesure de probabilité sur {0,1,2,...} définie par
λ M (k) = #{a ∈ V M : d(o,a) = k}
, k ≥ 0.
#V M
Si M a n faces, on définit enfin le profil de M changé d’échelle par la relation suivante,
où A est un borélien de R + .
λ (n)
M (A) = λ M
(
n 1/4 A
)
,
Théorème 1.6.1. Soit q une suite de poids critique régulière.
(i) La loi de n −1/4 R M sous la mesure de probabilité B r q(· | #F M = n) converge quand
n → ∞ vers la loi sous la mesure N 0 de la variable aléatoire
(
)
1
sup Ŵ(s) − inf Ŵ(s) .
β q 0≤s≤1
0≤s≤1
(ii) La loi de la mesure aléatoire λ (n)
M sous la mesure de probabilité Br q (· | #F M = n)
converge quand n → ∞ vers la loi sous la mesure N 0 de la mesure aléatoire I définie
par
〈I,g〉 =
∫ 1
0
( ( 1
g
β q
Ŵ(t) −
inf
0≤s≤1
))
Ŵ(s) dt.
(iii) La loi de n −1/4 d(o,a) où a est uniformément distribué sur V M , sous la mesure de
probabilité B r q(· | #F M = n), converge quand n → ∞ vers la loi sous la mesure N 0 de
la variable aléatoire ( )
1
sup Ŵ(s) .
β q
0≤s≤1
Le théorème 1.6.1 est une généralisation des résultats obtenus par Chassaing & Schaeffer [14]
qui traitent le cas particulier q = q 2 des quadrangulations (voir aussi le théorème 8.2 dans [39]).
Le théorème 1.6.1 est aussi très proche du théorème 3 de [43] qui a largement motivé notre
étude. La différence vient de ce que [43] considère des cartes enracinées et pointées, et étudie
les distances à partir du sommet distingué et non du sommet racine comme nous le faisons ici.
Signalons que dans le cas q = q 2 , la constante qui apparaît dans le théorème 1.6.1 vaut
( )
1 8 1/4
= .
β q2 9
La preuve du théorème 1.6.1 repose sur un théorème limite pour les mobiles bien étiquetés.
23

Théorème 1.6.2. Soit q une suite de poids critique régulière. La loi sous la mesure de
probabilité P µ q ,ν,1(· | #A 1 = n, U ≥ 1) de
( (αq ( )
C(2(#A − 1)t) βq V (2(#A − 1)t)
n
)0≤t≤1
1/2 ,
n
)0≤t≤1
1/4
converge (au sens de la convergence étroite des mesures sur C([0,1], R) 2 ) quand n → ∞ vers la
loi de (ζ,Ŵ) sous la mesure N 0.
Le théorème 1.6.1 se déduit du théorème 1.6.2 en utilisant les propriétés vérifiées par la
bijection de Bouttier, di Francesco & Guitter. En particulier on observe que la loi de R M sous
la mesure B r q coïncide avec la loi de la variable aléatoire
sup {U v : v ∈ A}
sous la mesure P µ q ,ν,1(· | #A 1 = n,U ≥ 1). Or le théorème 1.6.2 implique que la loi sous la
mesure P µ q ,ν,1(· | #A 1 = n, U ≥ 1) de
1
n
supU 1/4 v
v∈A
converge quand n → ∞ vers la loi sous N 0 de la variable aléatoire
1
sup Ŵ s ,
β q 0≤s≤1
c’est-à-dire d’après le théorème 1.4.2, la loi sous N 0 de la variable aléatoire
(
)
1
sup Ŵ s − inf Ŵ s .
β q 0≤s≤1
0≤s≤1
La preuve du théorème 1.6.2 s’articule de manière analogue à la preuve du théorème 2.2 dans
[39]. En particulier, une étape cruciale est le calcul d’estimations de la probabilité
P µ q ,ν,1
(
U ≥ 1 | #A 1 = n ) .
Proposition 1.6.3. Il existe des constantes γ > 0 et ˜γ > 0 telles que pour tout n suffisamment
grand,
γ
n ≤ P (
µ q ,ν,1 U ≥ 1 | #A 1 = n ) ≤ ˜γ n .
Cette proposition nous permet d’obtenir des résultats asymptotiques concernant les points
de disconnection dans les grandes 2κ-angulations enracinées uniformes.
Soit M une carte planaire et soit σ 0 un sommet de M. Soit σ ∈ V M \ {σ 0 }. On note S σ 0,σ
M
l’ensemble des sommets a de M tel que tout chemin allant de σ à a passe par σ 0 . On dit que σ 0
est un point de disconnection de M s’il existe σ ∈ V M \ {σ 0 } tel que S σ 0,σ
M ≠ {σ 0} et l’on note
D M l’ensemble des points de disconnection de la carte M.
Soit (A,U) un mobile et soit v ∈ A 0 . Rappelons que τ v A = {w ∈ U,vw ∈ A}. Supposons que
(τ v A) 0 ≠ {v} et que la condition suivante est satisfaite :
inf { U vw : w ∈ (τ v A) 0 \ {∅} } > U v .
On remarque alors que dans la construction de la carte M = Ψ r,p ((A,U)) l’ensemble S v,τ
M
est en
bijection avec (τ v A) 0 (où l’on a noté τ le point distingué de la carte enracinée et pointée M).
Le sommet v est donc un point de disconnection de M.
24

Rappelons que U n κ et U n κ désignent respectivement la mesure uniforme sur l’ensemble des
2κ-angulations enracinées et pointées à n faces et la mesure uniforme sur l’ensemble des 2κangulations
enracinées à n faces. On montre alors le résultat suivant en utilisant la remarque
précédente et la proposition 1.6.3.
Théorème 1.6.4. Pour tout ε > 0,
lim
n→∞ Un κ
(
∃ σ 0 ∈ D M : σ 0 ≠ τ, n 1/2−ε ≤ #S σ 0,τ
M ≤ 2n1/2−ε) = 1.
Remarquons que si M est une 2κ-angulation à n faces alors d’après la formule d’Euler
#V M = n(κ − 1) + 2.
On déduit alors du théorème 1.6.4 le résultat suivant.
Théorème 1.6.5. Pour tout ε > 0,
lim
n→∞ U n κ
(
∃ σ 0 ∈ D M : ∃ σ ∈ V M \ {σ 0 }, n 1/2−ε ≤ #S σ 0,σ
M ≤ 2n1/2−ε) = 1.
25

CHAPITRE 2
Regenerative real trees
2.1. Introduction
Galton-Watson trees are well known to be characterized among all random discrete trees by
a regenerative property. More precisely, if µ is a probability measure on Z + , the law Π µ of the
Galton-Watson tree with offspring distribution µ is uniquely determined by the following two
conditions : under the probability measure Π µ ,
(i) the ancestor has p children with probability µ(p),
(ii) if µ(p) > 0, then conditionally on the event that the ancestor has p children, the p subtrees
which describe the genealogy of the descendants of these children are independent
and distributed according to Π µ .
The aim of this work is to study σ-finite measures satisfying an analogue of this property on
the space of equivalence classes of rooted compact R-trees.
An R-tree is a metric space (T ,d) such that for any two points σ 1 and σ 2 in T there is a
unique arc with endpoints σ 1 and σ 2 , and furthermore this arc is isometric to a compact interval
of the real line. We denote this arc by [[σ 1 ,σ 2 ]]. In this work, all R-trees are supposed to be
compact. A rooted R-tree is an R-tree with a distinguished vertex called the root. Say that two
rooted R-trees are equivalent if there is a root-preserving isometry that maps one onto the other.
It was noted in [24] that the set T of all equivalence classes of rooted compact R-trees equipped
with the pointed Gromov-Hausdorff distanceGH (see e.g. Chapter 7 in [12]) is a Polish space.
Hence it is legitimate to consider random variables with values in T, that is, random R-trees.
A particularly important example is the CRT, which was introduced by Aldous [3], [5] with a
different formalism. Striking applications of the concept of random R-trees can be found in the
recent papers [24] and [25].
Let T be an R-tree. We write H(T ) for the height of the R-tree T , that is, the maximal
distance from the root to a vertex of T . For every t ≥ 0, we denote by T ≤t the set of all vertices
of T which are at distance at most t from the root, and by T >t the set of all vertices which are
at distance greater than t from the root. To each connected component of T >t there corresponds
a “subtree” of T above level t (see section 2.2.2.3 for a more precise definition). For every h > 0,
we define Z(t,t+h)(T ) as the number of subtrees of T above level t with height greater than h.
Let Θ be a σ-finite measure on T such that 0 < Θ(H(T ) > t) < ∞ for every t > 0 and
Θ(H(T ) = 0) = 0. We say that Θ satisfies the regenerative property (R) if the following holds :
(R) for every t,h > 0 and p ∈ N, under the probability measure Θ(· | H(T ) > t) and
conditionally on the event {Z(t,t + h) = p}, the p subtrees of T above level t with
height greater than h are independent and distributed according to the probability
measure Θ(· | H(T ) > h).
This is a natural analogue of the regenerative property stated above for Galton-Watson trees.
Beware that, unlike the discrete case, there is no natural order on the subtrees above a given
27

level. So, the preceding property should be understood in the sense that the unordered collection
of the p subtrees in consideration is distributed as the unordered collection of p independent
copies of Θ(· | H(T ) > h).
Property (R) is known to be satisfied by a wide class of infinite measures on T, namely the
“laws” of Lévy trees. Lévy trees have been introduced by T. Duquesne and J.F. Le Gall in
[22]. Their precise definition is recalled in section 2.2.3, but let us immediately give an informal
presentation.
Let Y be a critical or subcritical continuous-state branching process. The distribution of Y
is characterized by its branching mechanism function ψ. Assume that Y becomes extinct a.s.,
which is equivalent to the condition ∫ ∞
1
ψ(u) −1 du < ∞. The ψ-Lévy tree is a random variable
taking values in (T,GH), which describes the genealogy of a population evolving according to
Y and starting with infinitesimally small mass. More precisely, the “law” of the Lévy tree is
defined in [22] as a σ-finite measure on the space (T,GH), such that 0 < Θ ψ (H(T ) > t) < ∞
for every t > 0. As a consequence of Theorem 4.2 of [22], the measure Θ ψ satisfies Property
(R). In the special case ψ(u) = u α , 1 < α ≤ 2 corresponding to the so-called stable trees, this
was used by Miermont [46], [47] to introduce and to study certain fragmentation processes.
In the present work we describe all σ-finite measures on T that satisfy Property (R). We
show that the only infinite measures satisfying Property (R) are the measures Θ ψ associated
with Lévy trees. On the other hand, if Θ is a finite measure satisfying Property (R), we can
obviously restrict our attention to the case Θ(T) = 1 and we obtain that Θ must be the law of
the genealogical tree associated with a continuous-time discrete-state branching process.
Theorem 2.1.1. Let Θ be an infinite measure on (T,GH) such that Θ(H(T ) = 0) = 0 and
0 < Θ(H(T ) > t) < +∞ for every t > 0. Assume that Θ satisfies Property (R). Then, there
exists a continuous-state branching process, whose branching mechanism is denoted by ψ, which
becomes extinct almost surely, such that Θ = Θ ψ .
Theorem 2.1.2. Let Θ be a probability measure on (T,GH) such that Θ(H(T ) = 0) = 0
and 0 < Θ(H(T ) > t) for every t > 0. Assume that Θ satisfies Property (R). Then there
exists a > 0 and a critical or subcritical probability measure γ on Z + \{1} such that Θ is the
law of the genealogical tree for a discrete-space continuous-time branching process with offspring
distribution γ, where branchings occur at rate a.
In other words, Θ in Theorem 2.1.2 can be described in the following way : there exists a real
random variable J such that under Θ :
(i) J is distributed according to the exponential distribution with parameter a and there
exists σ J ∈ T such that T ≤J = [[ρ,σ J ]],
(ii) the number of subtrees above level J is distributed according to γ and is independent
of J,
(iii) for every p ≥ 2, conditionally on J and given the event that the number of subtrees
above level J is equal to p, these p subtrees are independent and distributed according
to Θ.
Theorem 2.1.1 is proved in section 2.3, after some preliminary results have been established in
section 2.2. A key idea of the proof is to use the regenerative property to embed discrete Galton-
Watson trees in our random real trees (Lemma 2.3.3). A technical difficulty comes from the
fact that real trees are not ordered whereas Galton-Watson trees are usually defined as random
ordered discrete trees (cf subsection 2.2.2.4 below). To overcome this difficulty, we assign a
28

andom ordering to the discrete trees embedded in real trees. Another major ingredient of the
proof of Theorem 2.1.1 is the construction of a “local time” L t at every level t of a random real
tree governed by Θ. The local time process is then shown to be a continuous-state branching
process with branching mechanism ψ, which makes it possible to identify Θ with Θ ψ . Theorem
2.1.2 is proved in section 2.4. Several arguments are similar to the proof of Theorem 2.1.1, so
that we have skipped some details.
2.2. Preliminaries
In this section, we recall some basic facts about branching processes, R-trees and Lévy trees.
2.2.1. Branching processes.
2.2.1.1. Continuous-state branching processes. A (continuous-time) continuous-state branching
process (in short a CSBP) is a Markov process Y = (Y t ,t ≥ 0) with values in the positive
half-line [0,+∞), with a Feller semigroup (Q t ,t ≥ 0) satisfying the following branching property :
for every t ≥ 0 and x,x ′ ≥ 0,
Q t (x, ·) ∗ Q t (x ′ , ·) = Q t (x + x ′ , ·).
Informally, this means that the union of two independent populations started respectively at x
and x ′ will evolve like a single population started at x + x ′ .
We will consider only the critical or subcritical case, meaning that for every t ≥ 0 and x ≥ 0,
∫
yQ t (x,dy) ≤ x.
[0,+∞)
Then, if we exclude the trivial case where Q t (x, ·) = δ 0 for every t > 0 and x ≥ 0, the Laplace
functional of the semigroup can be written in the following form : for every λ ≥ 0,
∫
e −λy Q t (x,dy) = exp(−xu(t,λ)),
[0,+∞)
where the function (u(t,λ),t ≥ 0,λ ≥ 0) is determined by the differential equation
du(t,λ)
= −ψ(u(t,λ)), u(0,λ) = λ,
dt
and ψ : R + −→ R + is of the form
∫
(2.2.1) ψ(u) = αu + βu 2 + (e −ur − 1 + ur)π(dr),
(0,+∞)
where α,β ≥ 0 and π is a σ-finite measure on (0,+∞) such that ∫ (0,+∞) (r ∧ r2 )π(dr) < ∞. The
process Y is called the ψ-continuous-state branching process (in short the ψ-CSBP).
Continuous-state branching processes may also be obtained as weak limits of rescaled Galton-
Watson processes. We recall that an offspring distribution is a probability measure on Z + . An
offspring distribution µ is said to be critical if ∑ i≥0 iµ(i) = 1 and subcritical if ∑ i≥0
iµ(i) < 1.
Let us state a result that can be derived from [26] and [30].
Theorem 2.2.1. Let (µ n ) n≥1 be a sequence of offspring distributions. For every n ≥ 1, denote
by X n a Galton-Watson process with offspring distribution µ n , started at X n 0 = n. Let (m n) n≥1
be a nondecreasing sequence of positive integers converging to infinity. We define a sequence of
processes (Y n ) n≥1 by setting, for every t ≥ 0 and n ≥ 1,
Y n
t = n −1 X n [m nt] .
29

Assume that, for every t ≥ 0, the sequence (Y n
t ) n≥1 converges in distribution to Y t where Y =
(Y t ,t ≥ 0) is an almost surely finite process such that P(Y δ > 0) > 0 for some δ > 0. Then, Y
is a continuous-state branching process and the sequence of processes (Y n ) n≥1 converges to Y in
distribution in the Skorokhod space D(R + ).
Proof : It follows from the proof of Theorem 1 of [30] that Y is a CSBP. Then, thanks to
Theorem 2 of [30], there exists a sequence of offspring distributions (ν n ) n≥1 and a nondecreasing
sequence of positive integers (c n ) n≥1 such that we can construct for every n ≥ 1 a Galton-Watson
process Z n started at c n and with offspring distribution ν n satisfying
where the symbol
(
c −1
n Zn [nt] ,t ≥ 0 )
d
−→
n→∞ (Y t,t ≥ 0),
d
−→ indicates convergence in distribution in D(R + ).
Let (m nk ) k≥1 be a strictly increasing subsequence of (m n ) n≥1 . For n ≥ 1, we set B n = X n k
and b n = n k if n = m nk for some k ≥ 1, and we set B n = Z n and b n = c n if there is no
k ≥ 1 such that n = m nk . Then, for every t ≥ 0, (b −1
n Bn [nt] ) n≥1 converges in distribution to Y t .
Applying Theorem 4.1 of [26], we obtain that
( )
b −1
n Bn [nt] ,t ≥ 0 d
−→ (Y t,t ≥ 0).
n→∞
In particular, we have,
( )
(2.2.2)
Y n k d
t ,t ≥ 0 −→ (Y t,t ≥ 0).
k→∞
As (2.2.2) holds for every strictly increasing subsequence of (m n ) n≥1 , we obtain the desired
result.
□
2.2.1.2. Discrete-state branching processes. A (continuous-time) discrete-state branching process
(in short DSBP) is a continuous-time Markov chain Y = (Y t ,t ≥ 0) with values in Z + whose
transition probabilities (P t (i,j),t ≥ 0) i≥0,j≥0 satisfy the following branching property : for every
i ∈ Z + , t ≥ 0 and |s| ≤ 1,
⎛ ⎞i
∞∑
∞∑
P t (i,j)s j = ⎝ P t (1,j)s j ⎠ .
j=0
We exclude the trivial case where P t (i,i) = 1 for every t ≥ 0 and i ∈ Z + . Then, there exist
a > 0 and an offspring distribution γ with γ(1) = 0 such that the generator of Y can be written
of the form ⎛
⎞
0 0 0 0 0 ...
aγ(0) −a aγ(2) aγ(3) aγ(4) ...
Q =
0 2aγ(0) −2a 2aγ(2) aγ(3) ...
.
⎜
⎝ 0 0 3aγ(0) −3a 3aγ(2) ... ⎟
⎠
.
. . .. . .. . .. . ..
Furthermore, it is well known that Y becomes extinct almost surely if and only if γ is critical
or subcritical. We refer the reader to [7] and [49] for more details.
2.2.2. Deterministic trees.
j=0
30

2.2.2.1. The space (T,GH) of rooted compact R-trees. We start with a basic definition.
Definition 2.2.1. A metric space (T ,d) is an R-tree if the following two properties hold for
every σ 1 ,σ 2 ∈ T .
(i) There is a unique isometric map f σ1 ,σ 2
from [0,d(σ 1 ,σ 2 )] into T such that f σ1 ,σ 2
(0) = σ 1
and f σ1 ,σ 2
(d(σ 1 ,σ 2 )) = σ 2 .
(ii) If q is a continuous injective map from [0,1] into T such that q(0) = σ 1 and q(1) = σ 2 ,
we have
q([0,1]) = f σ1 ,σ 2
([0,d(σ 1 ,σ 2 )]).
A rooted R-tree is an R-tree with a distinguished vertex ρ = ρ(T ) called the root.
In what follows, R-trees will always be rooted.
Let (T ,d) be an R-tree with root ρ, and σ,σ 1 ,σ 2 ∈ T . We write [[σ 1 ,σ 2 ]] for the range of the
map f σ1 ,σ 2
. In particular, [[ρ,σ]] is the path going from the root to σ and can be interpreted as
the ancestral line of σ.
The height H(T ) of the R-tree T is defined by H(T ) = sup{d(ρ,σ) : σ ∈ T }. In particular,
if T is compact, its height H(T ) is finite.
Two rooted R-trees T and T ′ are called equivalent if there is a root-preserving isometry that
maps T onto T ′ . We denote by T the set of all equivalence classes of rooted compact R-trees.
We often abuse notation and identify a rooted compact R-tree with its equivalence class.
The set T can be equipped with the pointed Gromov-Hausdorff distance, which is defined
as follows. If (E,δ) is a metric space, we use the notation δ Haus for the usual Hausdorff metric
between compact subsets of E. Then, if T and T ′ are two rooted compact R-trees with respective
roots ρ and ρ ′ , we define the distanceGH(T , T ′ ) as
{
GH(T , T ′ (
) = inf δ Haus φ(T ),φ ′ (T ′ ) ) }
∨ δ(φ(ρ),φ ′ (ρ ′ )) ,
where the infimum is over all isometric embeddings φ : T −→ E and φ ′ : T ′ −→ E into a
common metric space (E,δ). We see thatGH(T , T ′ ) only depends on the equivalence classes
of T and T ′ . Furthermore, according to Theorem 2 in [24],GH defines a metric on T that
makes it complete and separable. Notice thatGH(T , T ′ ) makes sense more generally if T and
T ′ are pointed compact metric spaces (see e.g. Chapter 7 in [12]). We will use this in the proof
of Lemma 2.2.2 below.
We equip T with its Borel σ-field.
If T ∈ T, we set T ≤t = {σ ∈ T : d(ρ,σ) ≤ t} for every t ≥ 0. Plainly, T ≤t is an R-tree whose
root is the same as the root of T . Note that the mapping T ↦−→ T ≤t from T into T is Lipschitz
for the Gromov-Hausdorff metric.
2.2.2.2. The R-tree coded by a function. We now recall a construction of rooted compact
R-trees which is described in [22]. Let g : [0,+∞) −→ [0,+∞) be a continuous function with
compact support satisfying g(0) = 0. We exclude the trivial case where g is identically zero. For
every s,t ≥ 0, we set
m g (s,t) = inf g(r)
r∈[s∧t,s∨t]
and
d g (s,t) = g(s) + g(t) − 2m g (s,t).
31

We define an equivalence relation ∼ on [0,+∞) by declaring that s ∼ t if and only if d g (s,t) = 0
(or equivalently if and only if g(s) = g(t) = m g (s,t)). Let T g be the quotient space
T g = [0,+∞)/ ∼ .
Then, d g induces a metric on T g and we keep the notation d g for this metric. According to
Theorem 2.1 of [22], the metric space (T g ,d g ) is a compact R-tree. By convention, its root is the
equivalence class of 0 for ∼ and is denoted by ρ g .
2.2.2.3. Subtrees of a tree above a fixed level. Let (T ,d) ∈ T and t > 0. Denote by T i,◦ ,i ∈ I
the connected components of the open set T >t = {σ ∈ T : d(ρ(T ),σ) > t}. Let i ∈ I. Then
the ancestor of σ at level t, that is, the unique vertex on the line segment [[ρ,σ]] at distance
t from ρ, must be the same for all σ ∈ T i,◦ . We denote by σ i this common ancestor and set
T i = T i,◦ ∪ {σ i }. Thus T i is a compact rooted R-tree with root σ i . The trees T i ,i ∈ I are called
the subtrees of T above level t. We now consider, for every h > 0,
Z(t,t + h)(T ) = #{i ∈ I : H(T i ) > h}.
By a compactness argument, we can easily verify that Z(t,t + h)(T ) < ∞.
2.2.2.4. Discrete trees and real trees. We start with some formalism for discrete trees. We
first introduce the set of labels
U = ⋃ n≥0
N n ,
where by convention N 0 = {∅}. An element of U is a sequence u = u 1 ...u n , and we set |u| = n
so that |u| represents the generation of u. In particular, |∅| = 0. If u = u 1 ...u n and v = v 1 ...v m
belong to U, we write uv = u 1 ...u n v 1 ...v m for the concatenation of u and v. In particular,
∅u = u∅ = u. The mapping π : U\{∅} −→ U is defined by π(u 1 ...u n ) = u 1 ... u n−1 (π(u) is
the father of u). Note that π k (u) = ∅ if k = |u|.
A rooted ordered tree θ is a finite subset of U such that
(i) ∅ ∈ θ,
(ii) u ∈ θ\{∅} ⇒ π(u) ∈ θ,
(iii) for every u ∈ θ, there exists a number k u (θ) ≥ 0 such that uj ∈ θ if and only if
1 ≤ j ≤ k u (θ).
We denote by A the set of all rooted ordered trees. If θ ∈ A, we write H(θ) for the height of θ, that
is H(θ) = max{|u| : u ∈ θ}. And for every u ∈ θ, we define τ u θ ∈ A by τ u θ = {v ∈ U : uv ∈ θ}.
This is the tree θ shifted at u.
Let us define an equivalence relation on A by setting θ ≡ θ ′ if and only if we can find a
permutation ϕ u of the set {1,... ,k u (θ)} for every u ∈ θ such that k u (θ) ≥ 1, in such a way that
{
θ ′ (
= {∅} ∪ ϕ ∅ u
1 ) (
ϕ u 1 u
2 ) ...ϕ u 1 ...u n−1 (un ) : u 1 ...u n ∈ θ, n ≥ 1}
.
In other words θ ≡ θ ′ if they correspond to the same unordered tree. Let A = A/ ≡ be the
associated quotient space and letÔ:A−→ A be the canonical projection. It is immediate that
if θ ≡ θ ′ , then k ∅ (θ) = k ∅ (θ ′ ). So, for every ξ ∈ A, we may define k ∅ (ξ) = k ∅ (θ) where θ is any
representative of ξ.
Let us fix ξ ∈ A such that k ∅ (ξ) = k > 0 and choose a representative θ of ξ. We may
define {ξ 1 ,... ,ξ k } = {Ô(τ 1 θ),...,Ô(τ k θ)} as the unordered family of subtrees of ξ above the
32

first generation. Then, if F : A k −→ R + is any symmetric measurable function, we have
( #Ô−1 (ξ) ) −1
∑
F(τ 1 θ,...,τ k θ)
(2.2.3)
=
θ∈Ô−1 (ξ)
( ) −1 ( ) −1 ∑
#Ô−1 (ξ 1 ) ... #Ô−1 (ξ k )
θ 1 ∈Ô−1 (ξ 1 )
...
∑
θ k ∈Ô−1 (ξ k )
F(θ 1 ,...,θ k ).
Note that the right-hand side of (2.2.3) is well defined since it is symmetric in {ξ 1 ,...,ξ k }. The
identity (2.2.3) is a simple combinatorial fact, whose proof is left to the reader.
A marked tree is a pair T = (θ, {h u } u∈θ ) where θ ∈ A and h u ≥ 0 for every u ∈ θ. We denote
by M the set of all marked trees. We can associate with every marked tree T = (θ, {h u } u∈θ ) ∈ M,
an R-tree T T in the following way. Let R θ be the vector space of all mappings from θ into R.
Write (e u ,u ∈ θ) for the canonical basis of R θ . We define l ∅ = 0 and l u = ∑ |u|
k=1 h π k (u)e π k (u) for
u ∈ θ \ {∅}. Let us set
T T =
u∈θ[l ⋃ u ,l u + h u e u ].
T T is a connected union of line segments in R θ . It is equipped with the distance d T such that
d T (a,b) is the length of the shortest path in T T between a and b, and can be rooted in ρ(T T ) = 0
so that it becomes a rooted compact R-tree.
If θ ∈ A, we write T θ for the R-tree T T where T = (θ, {h u } u∈θ ) with h ∅ = 0 and h u = 1
for every u ∈ θ \ {∅}, and we write d θ for the associated distance. We then set m ∅ = 0 and
m u = ∑ |u|−1
k=0 e π k (u) = l u + e u for every u ∈ θ \ {∅}.
It is easily checked that T θ = T θ′ if θ ≡ θ ′ . Thus for every ξ ∈ A, we may write T ξ for the
tree T θ where θ is any representative of ξ.
We will now explain how to approximate a general tree T in T by a discrete type tree. Let
ε > 0 and set T (ε) = {T ∈ T : H(T ) > ε}. For every T in T (ε) , we can construct by induction
an element ξ ε (T ) of A in the following way.
• If T ∈ T (ε) satisfies H(T ) ≤ 2ε, we set ξ ε (T ) =Ô({∅}).
• Let n be a positive integer. Assume that we have defined ξ ε (T ) for every T ∈ T (ε) such
that H(T ) ≤ (n + 1)ε. Let T be an R-tree such that (n + 1)ε < H(T ) ≤ (n + 2)ε. We
set k = Z(ε,2ε)(T ) and we denote by T 1 ,... , T k the k subtrees of T above level ε with
height greater than ε. Then ε < H(T i ) ≤ (n + 1)ε for every i ∈ {1,... ,k}, so we can
define ξ ε (T i ). Let us choose a representative θ i of ξ ε (T i ) for every i ∈ {1,... ,k}. We
set
ξ ε (T ) =Ô({∅} ∪ 1θ 1 ∪ ... ∪ kθ k) ,
where iθ i = {iu : u ∈ θ i }. Clearly this does not depend on the choice of the representatives
θ i .
If r > 0 and T is a compact rooted R-tree with metric d, we write rT for the same tree
equipped with the metric rd.
Lemma 2.2.2. For every ε > 0 and every T ∈ T (ε) , we have
( )
(2.2.4) GH εT ξε (T ) , T ≤ 4ε.
Proof : Let ε > 0 and T ∈ T. Let θ be any representative of ξ ε (T ). Recall the notation
(m u ,u ∈ θ). We can construct a mapping φ : θ −→ T such that :
33

(i) for every σ ∈ T , there exists u ∈ θ such that d(σ,φ(u)) ≤ 2ε,
(ii) for every u ∈ θ, d(φ(u),ρ) = ε|u| where ρ denotes the root of T ,
(iii) for every u,u ′ ∈ θ, 0 ≤ εd θ (m u ,m u ′) − d(φ(u),φ(u ′ )) ≤ 2ε.
To be specific, we always take φ(∅) = ρ, which suffices for the construction if H(T ) ≤ 2ε. If
(n + 1)ε < H(T ) ≤ (n + 2)ε for some n ≥ 1, we have as above
θ = {∅} ∪ 1θ 1 ∪ ... ∪ kθ k ,
where θ 1 ,... ,θ k are representatives of respectively ξ ε (T 1 ),... ,ξ ε (T k ), if T 1 ,... , T k are the
subtrees of T above level ε with height greater than ε. With an obvious notation we define
φ(ju) = φ j (u) for every j ∈ {1,... ,k} and u ∈ θ j . Properties (i)-(iii) are then easily checked by
induction.
Let us now set Φ = {φ(u),u ∈ θ} and M = {m u ,u ∈ θ}. We equip Φ with the metric
induced by d and M with the metric induced by εd θ . Then, Φ and M can be viewed as pointed
compact metric spaces with respective roots ρ and 0. It is immediate thatGH(εT θ ,M) ≤ ε.
Furthermore, Property (i) above implies thatGH(T ,Φ) ≤ 2ε. At last, according to Lemma 2.3
in [24] and Property (iii) above, we haveGH(Φ,M) ≤ ε. Lemma 2.2.2 then follows from the
triangle inequality forGH.
□
2.2.3. Lévy trees. Roughly speaking, a Lévy tree is a T-valued random variable which is
associated with a CSBP in such a way that it describes the genealogy of a population evolving
according to this CSBP.
2.2.3.1. The measure Θ ψ . We consider on a probability space (Ω,P) a ψ-CSBP Y = (Y t ,t ≥
0), where the function ψ is of the form (2.2.1), and we suppose that Y becomes extinct almost
surely. This condition is equivalent to
∫ ∞
du
(2.2.5)
ψ(u) < ∞.
This implies that at least one of the following two conditions holds :
∫
(2.2.6) β > 0 or rπ(dr) = ∞.
1
(0,1)
The Lévy tree associated to Y will be defined as the tree coded by the so-called height
process, which is a functional of the Lévy process with Laplace exponent ψ. Let us denote by
X = (X t ,t ≥ 0) a Lévy process on (Ω,P) with Laplace exponent ψ. This means that X is a
Lévy process with no negative jumps, and that for every λ,t ≥ 0,
E(exp(−λX t )) = exp(tψ(λ)).
Then, X does not drift to +∞ and has paths of infinite variation (by (2.2.6)).
We can define the height process H = (H t ,t ≥ 0) by the following approximation :
1
H t = lim
ε→0 ε
∫ t
0½{X s≤I s t +ε} ds,
where It s = inf{X r : s ≤ r ≤ t} and the convergence holds in probability (see Chapter 1 in [21]).
Informally, we can say that H measures the size of the set {s ∈ [0,t] : X s− ≤ It s }. Thanks to
condition (2.2.5), we know that the process H has a continuous modification (see Theorem 4.7
in [21]). From now on, we consider only this modification.
34

Let us now set I t = inf {X s : 0 ≤ s ≤ t} for every t ≥ 0, and consider the process X − I =
(X t −I t ,t ≥ 0). We recall that X −I is a strong Markov process for which the point 0 is regular.
The process −I is a local time for X − I at level 0. We write N for the associated excursion
measure. We let ∆(de) be the “law” of (H s ,s ≥ 0) under N. This makes sense because the
values of the height process in an excursion of X −I away from 0 only depend on that excursion
(see section 1.2 in [21]). Then, ∆(de) is a σ-finite measure on C([0, ∞)), and is supported on
functions with compact support such that e(0) = 0.
The Lévy tree is the tree (T e ,d e ) coded by the function e, in the sense of section 2.2.2.2,
under the measure ∆(de). We denote by Θ ψ the σ-finite measure on T which is the “law”of the
Lévy tree, that is the image of ∆(de) under the mapping e ↦−→ T e .
2.2.3.2. A discrete approximation of the Lévy tree. Let us now recall that the Lévy tree is
the limit in the Gromov-Hausdorff distance of suitably rescaled Galton-Watson trees.
We start by recalling the definition of Galton-Watson trees which was given informally in
the introduction above. Let µ be a critical or subcritical offspring distribution. We exclude the
trivial case where µ(1) = 1. Then, there exists a unique probability measure Π µ on A such that :
(i) for every p ≥ 0, Π µ (k ∅ = p) = µ(p),
(ii) for every p ≥ 1 with µ(p) > 0, under the probability measure Π µ (· | k ∅ = p), the shifted
trees τ 1 θ,...,τ p θ are independent and distributed according to Π µ .
Recall that if r > 0 and T is a compact rooted R-tree with metric d, we write rT for the
same tree equipped with the metric rd. The following result is Theorem 4.1 in [22].
Theorem 2.2.3. Let (µ n ) n≥1 be a sequence of critical or subcritical offspring distributions.
For every n ≥ 1, let us denote by X n a Galton-Watson process with offspring distribution µ n ,
started at X n 0 = n. Let (m n) n≥1 be a nondecreasing sequence of positive integers converging to
infinity. We define a sequence of processes (Y n ) n≥1 by setting, for every t ≥ 0 and n ≥ 1,
Y n
t = n −1 X n [m nt] .
Assume that, for every t ≥ 0, the sequence (Yt n ) n≥1 converges in distribution to Y t where Y =
(Y t ,t ≥ 0) is a ψ-CSBP which becomes extinct almost surely. Assume furthermore that for every
δ > 0,
lim inf P(Y n
n→∞
δ = 0) > 0.
Then, for every a > 0, the law of the R-tree m −1
n T θ under Π µn (· | H(θ) ≥ [am n ]) converges
as n → ∞ to the probability measure Θ ψ (· | H(T ) > a) in the sense of weak convergence of
measures in the space T.
2.3. Proof of Theorem 2.1.1
Let Θ be an infinite measure on (T,GH) satisfying the assumptions of Theorem 1.1. Clearly
Θ is σ-finite.
We start with two important lemmas that will be used throughout this section. Let us first
define v : (0, ∞) −→ (0, ∞) by v(t) = Θ(H(T ) > t) for every t > 0. For every t > 0, we denote
by Θ t the probability measure Θ(· | H(T ) > t).
Lemma 2.3.1. The function v is nonincreasing, continuous and verifies
v(t)
v(t)
−→ ∞,
t→0
−→ 0.
t→∞
35

Proof : We only have to prove the continuity of v. To this end, we argue by contradiction
and assume that there exists t > 0 such that Θ(H(T ) = t) > 0. Let s > 0 and u ∈ (0,t) such
that v(u) > v(t). From the regenerative property (R), we have
Θ s (H(T ) = s + t) = Θ s( )
Θ s (H(T ) = s + t | Z(s,s + u))
= Θ s ( Z(s,s + u)Θ u (H(T ) = t)(Θ u (H(T ) ≤ t)) Z(s,s+u)−1)
=
> 0.
Θ(H(T ) = t)
v(u)
(
Θ
(Z(s,s s + u) 1 − v(t) ) )
Z(s,s+u)−1
v(u)
We have shown that Θ(H(T ) = t + s) > 0 for every s > 0. This is absurd since Θ is σ-finite. □
Lemma 2.3.2. For every t > 0 and 0 < a < b, the conditional law of the random variable
Z(t,t+b), under the probability measure Θ t and given Z(t,t+a), is a binomial distribution with
parameters Z(t,t+a) and v(b)/v(a) (where we define the binomial distribution with parameters
0 and p ∈ [0,1] as the Dirac measure δ 0 ).
Proof : This is a straightforward consequence of the regenerative property.
□
2.3.1. The CSBP derived from Θ. In this section, we consider a random forest of trees
derived from a Poisson point measure with intensity Θ. We associate with this forest a family
of Galton-Watson processes. We then construct local times at every level a > 0 as limits of the
rescaled Galton-Watson processes. Finally we show that the local time process is a CSBP.
Let us now fix the framework. We consider a probability space (Ω, P) and on this space a
Poisson point measure N = ∑ i∈I δ T i
on T, whose intensity is the measure Θ.
2.3.1.1. A family of Galton-Watson trees. We start with some notation that we need in
the first lemma. We consider on another probability space (Ω ′ , P ′ ), a collection (θ ξ ,ξ ∈ A) of
independent A-valued random variables such that for every ξ ∈ A, θ ξ is distributed uniformly
overÔ−1 (ξ). In what follows, to simplify notation, we identify an element ξ of the set A with
the subsetÔ−1 (ξ) of A.
Recall the definition of ξ ε (T ) before Lemma 2.2.2.
Lemma 2.3.3. Let us define for every ε > 0, a mapping θ (ε) from T (ε) × Ω ′ into A by
θ (ε) (T ,ω) = θ ξ ε (T )(ω).
Then for every positive integer p, the law of the random variable θ (ε) under the probability
measure Θ pε ⊗ P ′ is Π µε (· | H(θ) ≥ p − 1) where µ ε denotes the law of Z(ε,2ε) under Θ ε .
Proof : Since {H(T ) > pε} × Ω ′ = {H(θ (ε) ) ≥ p − 1} for every p ≥ 1, it suffices to show
the result for p = 1. Let k be a nonnegative integer. According to the construction of ξ ε (T ), we
have
( Θ ε ⊗ P ′ k ∅
(θ (ε)) )
= k = Θ ε (Z(ε,2ε) = k) = µ ε (k).
36

Let us fix k ≥ 1 with µ ε (k) > 0. Let F : A k −→ R + be a symmetric measurable function. Then
we have
( Θ ε ⊗ P ′ F
(τ 1 θ (ε) ,...,τ k θ (ε)) ∣ (
∣ k∅ θ (ε)) )
= k
⎛
= Θ ε ⊗ P ′ ⎝ ∑
⎞
F (τ 1 θ,...,τ k θ)½{θ ξ ε (T ) =θ}
∣ Z(ε,2ε) = k ⎠
(2.3.1)
θ∈ξ ε (T )
= Θ ε ⎛
⎝(#ξ ε (T )) −1
∑
θ∈ξ ε (T )
F (τ 1 θ,...,τ k θ)
⎞
∣ Z(ε,2ε) = k ⎠ .
On the event {Z(ε,2ε) = k}, we write T 1 ,...,T k for the k subtrees of T above level ε with
height greater than ε. Then, Formula (2.2.3) and the regenerative property yield
⎛
⎞
∑
Θ ε ⎝(#ξ ε (T )) −1
F (τ 1 θ,...,τ k θ)
∣ Z(ε,2ε) = k ⎠
θ∈ξ ε (T )
= Θ ε ⎛
⎝(#ξ ε (T 1 )) −1 ...(#ξ ε (T k )) −1
=
=
∫
∫
∑
θ 1 ∈ξ ε (T 1 )
...
Θ ε (dT 1 )... Θ ε (dT k )(#ξ ε (T 1 )) −1 ...(#ξ ε (T k )) −1
∑
θ k ∈ξ ε (T k )
∑
θ 1 ∈ξ ε (T 1 )
⎞
F (θ 1 ,... ,θ k )
∣ Z(ε,2ε) = k ⎠
...
∑
θ k ∈ξ ε (T k )
Θ ε ⊗ P ′ (dT 1 ,dω ′ 1 )... Θε ⊗ P ′ (dT k ,dω ′ k )F (
θ (ε) (T 1 ,ω ′ 1 ),... ,θ(ε) (T k ,ω ′ k ) )
,
F (θ 1 ,... ,θ k )
as in (2.3.1). We ( have thus proved that
Θ ε ⊗ P ′ F
(τ 1 θ (ε) ,... ,τ k θ (ε))∣ (
∣ k∅ θ (ε)) )
= k
∫
(
)
(2.3.2) = Θ ε ⊗ P ′ (dT 1 ,dω 1 ′ )... Θε ⊗ P ′ (dT k ,dω k ′ )F θ (ε) (T 1 ,ω 1 ′ ),...,θ(ε) (T k ,ω k ′ ) .
Note that for every permutation ϕ of the set {1,... ,k}, the k-tuples (τ ϕ(1) θ (ε) ,...,τ ϕ(k) θ (ε) ) and
(τ 1 θ (ε) ,...,τ k θ (ε) ) have the same distribution under Θ ε ⊗P ′ . Then, (2.3.2) means that the law of
θ (ε) under Θ ε ⊗ P ′ satisfies the branching property of the Galton-Watson trees. This completes
the proof of the desired result.
□
Recall that ∑ i∈I δ T i
is a Poisson point measure on T with intensity Θ. Let us now set, for
every t,h > 0,
Z(t,t + h) = ∑ Z(t,t + h)(T i ).
i∈I
For every ε > 0, we define a process X ε = (Xk ε ,k ≥ 0) on (Ω, P) by the formula
= Z(kε,(k + 1)ε), k ≥ 0.
X ε k
Proposition 2.3.4. For every ε > 0, the process X ε is a Galton-Watson process whose initial
distribution is the Poisson distribution with parameter v(ε) and whose offspring distribution is
µ ε .
Proof : First note that X0 ε = N(H(T ) > ε) is Poisson with parameter Θ(H(T ) > ε) = v(ε).
Then let p be a positive integer. We know from a classical property of Poisson measures that,
37

under the probability measure P and conditionally on the event {X0 ε = p}, the atoms of N that
belong to the set T ε are distributed as p i.i.d. variables with distribution Θ ε . Furthermore, it
follows from Lemma 2.3.3 that under Θ ε , the process (Z(kε,(k + 1)ε)) k≥0 is a Galton-Watson
process started at one with offspring distribution µ ε . This completes the proof. □
As a consequence, we get the next proposition, which we will use throughout this work.
Proposition 2.3.5. For every t > 0 and h > 0, we have Θ(Z(t,t + h)) ≤ v(h).
Proof : Since compact R-trees have finite height, the Galton-Watson process X ε dies out P
a.s. This implies that µ ε is critical or subcritical so that (Xk ε ,k ≥ 0) is a supermartingale. Let
t,h > 0. We can find ε > 0 and k ∈ N such that t = kε and ε ≤ h. Thus we have,
(2.3.3) Θ(Z(t,t + ε)) = Θ(Z(kε,(k + 1)ε)) = E(X ε k ) ≤ E(X ε 0) = v(ε).
Using Lemma 2.3.2 and (2.3.3), we get
2.3.1.2. A local time process.
Θ(Z(t,t + h)) = Θ
(
Z(t,t + ε) v(h) )
≤ v(h).
v(ε)
Proposition 2.3.6. For every t ≥ 0, there exists a random variable L t on the space T such
that Θ a.e.,
Z(t,t + h)
v(h)
−→ L t.
h→0
Proof : Let us start with the case t = 0. As Z(0,h) =½{H(T )>h} for every h > 0, Lemma
2.3.1 gives v(h) −1 Z(0,h) → 0 Θ a.e. as h → 0, so we set L 0 = 0.
Let us now fix t > 0. Thanks to Lemma 2.3.1, we can define a decreasing sequence (ε n ) n≥1
by the condition v(ε n ) = n 4 for every n ≥ 1. We claim that there exists a random variable L t
on the space T such that, Θ a.e.,
Z(t,t + ε n )
(2.3.4)
n 4 −→ L t .
n→∞
Indeed, using Lemma 2.3.2, we have, for every n ≥ 1,
( ∣∣∣∣
Θ t Z(t,t + ε n )
n 4 − Z(t,t + ε ∣ )
n+1) ∣∣∣
2
(n + 1) 4
( ( ∣∣∣∣
= Θ t 1
n 4
n 8Θt Z(t,t + ε n ) −
(n + 1) 4Z(t,t + ε ∣ ))
n+1) ∣ ∣∣∣
∣2
Z(t,t + ε n+1 )
( )
Z(t,t +
≤ Θ t εn+1 )
4n 8
(2.3.5)
≤
(n + 1)4
4v(t)n 8 ,
where the last bound follows from Proposition 2.3.5 and the definition of ε n+1 . Thanks to the
Cauchy-Schwarz inequality, we get
(2.3.6) Θ t (∣ ∣∣∣ Z(t,t + ε n )
n 4
− Z(t,t + ε ∣)
n+1) ∣∣∣
(n + 1) 4 ≤
(n + 1)2
2n 4√ v(t) ≤ 2
n 2√ v(t) .
□
38

The bound (2.3.6) implies
(
∑ ∞ Θ
Z(t,t + ε n )
∣ n 4
In particular, Θ a.e.,
Our claim (2.3.4) follows.
n=1
∞∑
Z(t,t + ε n )
∣ n 4
n=1
− Z(t,t + ε ∣)
n+1) ∣∣∣
(n + 1) 4 < ∞.
− Z(t,t + ε n+1)
(n + 1) 4 ∣ ∣∣∣
< ∞.
For every h ∈ (0,ε 1 ], we can find n ≥ 1 such that ε n+1 ≤ h ≤ ε n . Then, we have Z(t,t+ε n ) ≤
Z(t,t + h) ≤ Z(t,t + ε n+1 ) Θ a.e., and n 4 ≤ v(h) ≤ (n + 1) 4 so that
We then deduce from (2.3.4) that Θ a.e.,
which completes the proof.
Z(t,t + ε n ) Z(t,t + h)
(n + 1) 4 ≤ ≤ Z(t,t + ε n+1)
v(h) n 4 .
Z(t,t + h)
v(h)
−→
h→0 L t
Definition 2.3.1. We define a process L = (L t ,t ≥ 0) on (Ω, P) by setting L 0 = 1 and for
every t > 0,
L t = ∑ L t (T i ).
i∈I
□
Notice that L t (T ) = 0 if H(T ) ≤ t so that the above sum is finite a.s.
Corollary 2.3.7. For every t ≥ 0, we have P a.s.
Z(t,t + h)
v(h)
−→ L t.
h→0
Moreover, this convergence holds in L 1 (P) uniformly in t ∈ [0, ∞).
Proof : The first assertion is an immediate consequence of Proposition 2.3.6. Let us focus
on the second assertion. From Lemma 2.3.2 we have
(
)
Θ n −4 Z(t,t + ε n ) − (n + 1) −4 Z(t,t + ε n+1 ) = 0
for every t ≥ 0 and n ≥ 1. Thus, from the second moment formula for Poisson measures, we get,
for every t ≥ 0 and n ≥ 1,
( (Z(t,t + εn )
E
n 4 − Z(t,t + ε ) ) ( 2 (Z(t,t
n+1)
+ εn )
(n + 1) 4 = Θ
n 4 − Z(t,t + ε ) ) 2
n+1)
(n + 1) 4 .
Now, we have
( (Z(0,εn
)
Θ
n 4
− Z(0,ε ) ) ( 2 ) ) 2
n+1)
(½{H(T )>ε
(n + 1) 4 = Θ
n} )>ε n+1 }
−½{H(T
n 4 (n + 1) 4 = 1 n 4 − 1
(n + 1) 4
and for every t > 0, thanks to the bound (2.3.5),
( (Z(t,t + εn )
Θ
n 4 − Z(t,t + ε ) ) 2
n+1) (n + 1)4
(n + 1) 4 ≤
4n 8 .
39

So for every t ≥ 0 and n ≥ 1, we have from the Cauchy-Schwarz inequality
(∣ ∣∣∣ Z(t,t + ε n )
(2.3.7) E
n 4 − Z(t,t + ε ∣)
n+1) ∣∣∣ (n + 1)2
(n + 1) 4 ≤
n 4 .
Then n −4 Z(t,t + ε n ) → L t in L 1 as n → ∞ and, for every n ≥ 2,
(∣ ∣)
∣∣∣ Z(t,t + ε n ) ∣∣∣
∞∑ (k + 1) 2 ∞∑ 4
E
n 4 − L t ≤
k 4 ≤
k 2 ≤ 8 n .
k=n
In the same way as in the proof of (2.3.7), we have the following inequality : if h ∈ (0,ε 1 ], t ≥ 0
and n is a positive integer such that ε n+1 ≤ h ≤ ε n ,
(∣ ) √ ∣∣∣ Z(t,t + ε n ) Z(t,t + h)
E
n 4 − v(h)
v(h) ∣ ≤
n 4 ≤ √ 16 .
v(h)
Then, for every h ∈ (0,ε 2 ] and t ≥ 0, we get
(∣ ∣) ∣∣∣ Z(t,t + h) ∣∣∣ (
E
− L t ≤ 16 v(h) −1/2 + v(h) −1/4) ,
v(h)
which completes the proof.
We will now establish a regularity property of the process (L t ,t ≥ 0).
Proposition 2.3.8. The process (L t ,t ≥ 0) admits a modification, denoted by ( ˜L t ,t ≥ 0),
which is right-continuous with left-limits, and which has no fixed discontinuities.
Proof : We start with two lemmas.
Lemma 2.3.9. There exists λ ≥ 0 such that E(L t ) = e −λt for every t ≥ 0.
Proof : We claim that the function t ∈ [0,+∞) ↦−→ E(L t ) is multiplicative, meaning that
for every t,s ≥ 0, E(L t+s ) = E(L t )E(L s ). As L 0 = 1 by definition, E(L 0 ) = 1. Let t,s > 0 and
0 < h < s. Let us denote by T 1 ,... , T Z(t,t+h) the subtrees of T above level t with height greater
than h. Then, using the regenerative property, we can write
⎛
Θ(Z(t + s,t + s + h)) = Θ ⎝
Z(t,t+h)
∑
i=1
⎞
k=n
(
)
Z(s,s + h)(T i ) ⎠ = Θ Z(t,t + h)Θ h (Z(s,s + h)) ,
which implies
( )
Z(s,s + h)
(2.3.8) E(Z(t + s,t + s + h)) = E(Z(t,t + h))E
.
v(h)
Thus, dividing by v(h) and letting h → 0 in (2.3.8), we get our claim from Corollary 2.3.7.
Moreover, thanks to Proposition 2.3.5 and Corollary 2.3.7, we know that E(L t ) ≤ 1 for every
t ≥ 0. Then, we obtain in particular that the function t ∈ [0, ∞) ↦−→ E(L t ) is nonincreasing.
To complete the proof, we have to check that E(L t ) > 0 for every t > 0. If we assume that
E(L t ) = 0 for some t > 0 then L t = 0, Θ a.e. Let s,h > 0 such that 0 < h < s. With the same
notation as in the beginning of the proof, we can write
Θ(H(T ) > t + s) = Θ ( ∃i ∈ {1,... ,Z(t,t + h)} : H(T i ) > s )
( (
= Θ 1 − 1 − v(s) ) ) Z(t,t+h)
(2.3.9)
.
v(h)
□
40

Now, thanks to Proposition 2.3.6, Θ a.e.,
(
1 − v(s) ) Z(t,t+h)
−→ exp(−L t v(s)) = 1.
v(h) h→0
Moreover, Θ a.e.,
(
1 − 1 − v(s) ) Z(t,t+h)
≤½{H(T
v(h)
)>t}.
Then, using dominated convergence in (2.3.9) as h → 0, we obtain Θ(H(T ) > t + s) = 0 which
contradicts the assumptions of Theorem 2.1.1.
□
Lemma 2.3.10. Let us denote by D = {k2 −n ,k ≥ 1,n ≥ 0} the set of positive dyadic numbers
and define G t = σ(L s ,s ∈ D,s ≤ t) for every t ∈ D. Then (L t ,t ∈ D) is a nonnegative
supermartingale with respect to the filtration (G t ,t ∈ D).
Proof : Let p be a positive integer, let s 1 ,... ,s p ,s,t ∈ D such that s 1 < ... < s p ≤ s < t
and let f : R p −→ R + be a bounded continuous function. We can find a positive integer n
such that 2 n t, 2 n s, and 2 n s i for i ∈ {1,... ,p} are nonnegative integers. The process X 2−n is a
subcritical Galton-Watson process, so
E
(
X 2−n
2 n t f
(
X 2−n
2 n s 1
,... , X 2−n
2 n s p
))
≤ E
( (
))
X2 2−n
n s f X2 2−n
n s 1
,... , X2 2−n
n s p
.
Therefore we have also,
( Z(t,t + 2 −n (
) Z(s1 ,s 1 + 2 −n )
E
v(2 −n f
) v(2 −n ,..., Z(s p,s p + 2 −n ))
)
) v(2 −n )
( Z(s,s + 2 −n (
) Z(s1 ,s 1 + 2 −n )
≤ E
v(2 −n f
) v(2 −n ,..., Z(s p,s p + 2 −n ))
)
(2.3.10)
) v(2 −n .
)
We can then use Corollary 2.3.7 to obtain
E ( L t f ( L s1 ,...,L sp
))
≤ E
(
Ls f ( L s1 ,... , L sp
))
.
We now complete the proof of Proposition 2.3.8. Let us set, for every t ≥ 0,
˜G t =
⋂
G s .
s>t,s∈D
From Lemma 2.3.10 and classical results on supermartingales, we can define a rightcontinuous
supermartingale ( ˜L t ,t ≥ 0) with respect to the filtration (˜G t ,t ≥ 0) by setting,
for every t ≥ 0,
(2.3.11) ˜Lt = lim
s↓t,s∈D L s,
where the limit holds P a.s. and in L 1 (see e.g. Chapter VI in [16] for more details). We claim
that ( ˜L t ,t ≥ 0) is a càdlàg modification of (L t ,t ≥ 0) with no fixed discontinuities.
We first prove that ( ˜L t ,t ≥ 0) is a modification of (L t ,t ≥ 0). For every t ≥ 0 and every
sequence (s n ) n≥0 in D such that s n ↓ t as n ↑ ∞, we have thanks to (2.3.11) and Lemma 2.3.9,
( )
E ˜Lt = lim E(L s
n→∞ n
) = E(L t ).
□
41

Let us now show that for every t ≥ 0, L t ≤ ˜L t P a.s. Let α,ε > 0 and δ ∈ (0,1). Thanks to
Corollary 2.3.7, we can find h 0 > 0 such that for every h ∈ (0,h 0 ) and n ≥ 0,
(∣ ∣) ∣∣∣ Z(t,t + h) ∣∣∣
E
− L t ≤ εα,
v(h)
(∣ ∣) ∣∣∣ Z(s n ,s n + h) ∣∣∣
E
− L sn ≤ εα.
v(h)
We choose h ∈ (0,h 0 ) and n 0 ≥ 0 such that s n − t + h ≤ h 0 and v(h) ≤ (1 + δ)v(s n − t + h) for
every n ≥ n 0 . We notice that Z(t,s n + h) ≤ Z(s n ,s n + h) so that, for every n ≥ n 0 ,
P(L t > (1 + δ)L sn + ε)
(
≤ P L t − Z(t,s n + h)
v(s n − t + h) > (1 + δ)L s n
− (1 + δ) Z(s )
n,s n + h)
+ ε
v(h)
(∣ )
(∣ ∣)
∣∣∣
≤ 2ε −1 Z(t,s n + h)
E
v(s n − t + h) − L ∣∣∣ t∣
+ 2ε −1 Z(s n ,s n + h) ∣∣∣
(1 + δ)E
− L sn
v(h)
≤ 6α.
We have thus shown that
(2.3.12) P(L t > (1 + δ)L sn + ε) −→
n→∞
0.
So, P(L t − (1 + δ) ˜L t > ε) = 0 for every ε > 0, implying that L t ≤ (1 + δ) ˜L t , P a.s. This leads
us to the claim L t ≤ ˜L t a.s. Since we saw that E(L t ) = E( ˜L t ), we have L t = ˜L t P a.s. for every
t ≥ 0.
Now, ( ˜L t ,t ≥ 0) is a right-continuous supermartingale. Thus, ( ˜L t ,t ≥ 0) is also left-limited
and we have E( ˜L t ) ≤ E( ˜L t− ) for every t > 0. Moreover, we can prove in the same way as we did
for (2.3.12) that, for every t > 0 and every sequence (s n ,n ≥ 0) in D such that s n ↑ t as n ↑ ∞,
(
P ˜Lsn > (1 + δ) ˜L
)
t + ε
−→ 0,
n→∞
implying that ˜L t− ≤ ˜L t , P a.s. So, L t = L t− P a.s. for every t > 0 meaning that ( ˜L t ,t ≥ 0) has
no fixed discontinuities.
□
From now on, to simplify notation, we replace (L t ,t ≥ 0) by its càdlàg modification ( ˜L t ,t ≥ 0).
2.3.1.3. The CSBP. We will prove that the suitably rescaled family of Galton-Watson processes
(X ε ) ε>0 converges to the local time L.
Thanks to Lemma 2.3.1, we can define a sequence (η n ) n≥1 by the condition v(η n ) = n for every
n ≥ 1. We set m n = [ηn −1 ] where [x] denotes the integer part of x. We recall from Proposition
2.3.4 that X ηn is a Galton-Watson process on (Ω, P) whose initial distribution is the Poisson
distribution with parameter n. For every n ≥ 1, we define a process Y n = (Yt n ,t ≥ 0) on (Ω, P)
by the following formula,
Yt n = n −1 X ηn
[m , t ≥ 0.
nt]
Proposition 2.3.11. For every t ≥ 0, Y n t → L t in probability as n → ∞.
Proof : Let t ≥ 0 and let δ > 0. We can write
P(|Yt n − L t| > 2δ) ≤ P (∣ ) (∣ ∣ )
∣ Y
n
t − L ηn[m
∣ nt] > δ + P ∣Lηn[m nt] − L t > δ
≤ δ −1 E (∣ ∣ Y
n
t − L ηn[m
∣ ) nt] + P (∣ ∣ )
∣ Lηn[mnt] − L t > δ .
42

Now, Corollary 2.3.7 and Proposition 2.3.8 imply respectively that
E (∣ ∣Yt n − L ∣ ) η n[m nt] −→
which completes the proof.
n→∞
0,
P (∣ ∣L ηn[m nt] − L t
∣ ∣ > δ ) −→
n→∞
0,
Corollary 2.3.12. For every t ≥ 0, the law of Y n t under P(· | X ηn
0 = n) converges weakly
to the law of L t under P as n → ∞.
Proof : For positive integers n and k, we denote by µ n the offspring distribution of the
Galton-Watson process X ηn , by f n the generating function of µ n and by fk n the k-th iterate
fk n = fn ◦ ... ◦ f n of f n . Let λ > 0 and t ≥ 0. We have,
∞
E(exp(−λYt n )) = ∑ (
e −nnp f[m n
p! nt]
(e −λ/n)) p ( ( (
= exp −n 1 − f[m n
nt]
e −λ/n))) .
p=0
From Proposition 2.3.11, it holds that
( ( (
exp −n 1 − f[m n
nt]
e −λ/n))) −→ E(exp(−λL t )).
n→∞
Let us set u(t,λ) = − log(E[exp(−λL t )]). It follows that,
( (
n 1 − f[m n
nt]
e −λ/n)) −→ u(t,λ).
n→∞
Thus, we obtain,
(
)
E exp(−λYt n ) ∣ X ηn
0 = n =
( (
f[m n
nt] e −λ/n)) n
−→ exp(−u(t,λ)) = E[exp(−λL t )].
n→∞
At this point, we can use Theorem 2.2.1 to assert that (L t ,t ≥ 0) is a CSBP and that the law
of (Yt n ,t ≥ 0) under the probability measure P(· | X ηn
0 = n) converges to the law of (L t ,t ≥ 0)
as n → ∞ in the space of probability measures on the Skorokhod space D(R + ). To verify the
assumptions of Theorem 2.2.1, we need to check that there exists δ > 0 such that P(L δ > 0) > 0.
This is obvious from Lemma 2.3.9.
2.3.2. Identification of the measure Θ. In the previous section, we have constructed
from Θ a CSBP L, which becomes extinct almost surely. We denote by ψ the associated branching
mechanism. We can consider the σ-finite measure Θ ψ , which is the law of the Lévy tree associated
with L. Our goal is to show that the measures Θ and Θ ψ coincide.
Recall that µ n denotes the offspring distribution of the Galton-Watson process X ηn .
Lemma 2.3.13. For every a > 0, the law of the R-tree η n T θ under Π µ n(· | H(θ) ≥ [am n ])
converges as n → ∞ to the probability measure Θ ψ (· | H(T ) > a) in the sense of weak convergence
of measures in the space T.
Proof : We first check that, for every δ > 0,
(2.3.13) lim inf
n→∞ P(Yn δ = 0) > 0.
Indeed, we have
) (
)
P(Yδ (N(H(T n = 0) = P ) > ([m n δ] + 1)η n ) = 0 = exp − v(([m n δ] + 1)η n ) .
□
□
43

As v is continuous, it follows that P(Yδ n = 0) → exp(−v(δ)) as n → ∞ implying (2.3.13).
We recall that the law of Y n under the probability measure P(· | X ηn
0 = n) converges to the
law of (L t ,t ≥ 0). Then, thanks to (2.3.13), we can apply Theorem 2.2.3 to get that, for every
a > 0, the law of the R-tree m −1
n T θ under Π µ n(· | H(θ) ≥ [am n ]) converges to the probability
measure Θ ψ (· | H(T ) > a) in the sense of weak convergence of measures in the space T. As
m −1
n η n → 1 as n → ∞, we get the desired result.
□
We can now complete the proof of Theorem 2.1.1. Indeed, thanks to Lemmas 2.2.2 and 2.3.3,
we can construct on the same probability space (Ω,P), a sequence of T-valued random variables
(T n ) n≥1 distributed according to Θ(· | H(T ) > ([am n ] + 1)η n ) and a sequence of A-valued
random variables (θ n ) n≥1 distributed according to Π µ n(· | H(θ) ≥ [am n ]) such that for every
n ≥ 1, P a.s., )
GH
(T n ,η n T θn ≤ 4η n .
Then, using Lemma 2.3.13, we have Θ(· | H(T ) > ([am n ]+1)η n ) → Θ ψ (· | H(T ) > a) as n → ∞
in the sense of weak convergence of measures on the space T. So we get
for every a > 0, and thus Θ = Θ ψ .
Θ(· |H(T ) > a) = Θ ψ (· |H(T ) > a)
2.4. Proof of Theorem 2.1.2
Let Θ be a probability measure on (T,GH) satisfying the assumptions of Theorem 2.1.2.
In this case, we define v : [0, ∞) −→ (0, ∞) by v(t) = Θ(H(T ) > t) for every t ≥ 0. Note
that v(0) = 1 is well defined here. For every t > 0, we denote by Θ t the probability measure
Θ(· | H(T ) > t). The following two results are proved in a similar way to Lemma 2.3.1 and
Lemma 2.3.2.
Lemma 2.4.1. The function v is nonincreasing, continuous and goes to 0 as t → ∞.
Lemma 2.4.2. For every t > 0 and 0 < a < b, the conditional law of the random variable
Z(t,t+b), under the probability measure Θ t and given Z(t,t+a), is a binomial distribution with
parameters Z(t,t + a) and v(b)/v(a).
2.4.1. The DSBP derived from Θ. We will follow the same strategy as in section 2.3
but instead of a CSBP we will now construct an integer-valued branching process.
2.4.1.1. A family of Galton-Watson trees. We recall that µ ε denotes the law of Z(ε,2ε) under
the probability measure Θ ε , and that (θ ξ ,ξ ∈ A) is a sequence of independent A-valued random
variables defined on a probability space (Ω ′ , P ′ ) such that for every ξ ∈ A, θ ξ is distributed
uniformly overÔ−1 (ξ). The following lemma is proved in the same way as Lemma 2.3.3.
Lemma 2.4.3. Let us define for every ε > 0, a mapping θ (ε) from T (ε) × Ω ′ into A by
θ (ε) (T ,ω) = θ ξ ε (T )(ω).
Then for every positive integer p, the law of the random variable θ (ε) under the probability
measure Θ pε ⊗ P ′ is Π µε (· | H(θ) ≥ p − 1).
For every ε > 0, we define a process X ε = (Xk ε ,k ≥ 0) on T by the formula
= Z(kε,(k + 1)ε), k ≥ 0.
X ε k
We show in the same way as Proposition 2.3.4 and Proposition 2.3.5 the following two results.
44

Proposition 2.4.4. For every ε > 0, the process X ε is under Θ a Galton-Watson process
whose initial distribution is the Bernoulli distribution with parameter v(ε) and whose offspring
distribution is µ ε .
Proposition 2.4.5. For every t > 0 and h > 0, we have Θ(Z(t,t + h)) ≤ v(h) ≤ 1.
The next proposition however is particular to the finite case and will be useful in the rest of
this section.
Proposition 2.4.6. The family of probability measures (µ ε ) ε>0 converges to the Dirac measure
δ 1 as ε → 0. In other words,
Proof : We first note that
Θ ε (Z(ε,2ε) = 1) −→
ε→0
1.
2Θ ε (Z(ε,2ε) ≥ 1) − Θ ε (Z(ε,2ε)) ≤ Θ ε (Z(ε,2ε) = 1) ≤ Θ ε (Z(ε,2ε) ≥ 1).
Moreover,
Θ ε (Z(ε,2ε) ≥ 1) = Θ ε (H(T ) > 2ε) = v(2ε)/v(ε)
and Θ ε (Z(ε,2ε)) ≤ 1. So,
(2.4.1)
2v(2ε)
v(ε)
− 1 ≤ Θ ε (Z(ε,2ε) = 1) ≤ v(2ε)
v(ε) .
We let ε → 0 in (2.4.1) and we use Lemma 2.4.1 to obtain the desired result.
2.4.1.2. Construction of the DSBP.
Proposition 2.4.7. For every t ≥ 0, there exists an integer-valued random variable L t on
the space T such that Θ(L t ) ≤ 1 and Θ a.s.,
Z(t,t + h) ↑ L t .
h↓0
Proof : Let t ≥ 0. The function h ∈ (0, ∞) ↦−→ Z(t,t + h) ∈ Z + is nonincreasing so that
there exists a random variable L t with values in Z + ∪ {∞} such that, Θ a.s.,
Z(t,t + h) ↑ L t .
h↓0
Thanks to the monotone convergence theorem, we have
Θ(Z(t,t + h)) −→
h↓0
Θ(L t ).
Now, by Proposition 2.4.5, Θ(Z(t,t + h)) ≤ 1 for every h > 0. Then, Θ(L t ) ≤ 1 which implies
in particular that L t < ∞ Θ a.s.
□
Proposition 2.4.8. For every t > 0, the following two convergences hold Θ a.s.,
(2.4.2)
(2.4.3)
Z(t − h,t)
Z(t − h,t + h)
↑
h↓0
↑
h↓0
L t ,
L t .
□
45

Proof : Let t > 0 be fixed throughout this proof. By the same arguments as in the proof
of Proposition 2.4.7, we can find a Z + -valued random variable L t such that Θ(L t ) ≤ 1 and
Z(t − h,t) ↑ L t as h ↓ 0, Θ a.s. If h ∈ (0,t), we write T 1 ,...,T Z(t−h,t) for the subtrees of T
above level t − h with height greater than h. Then, from the regenerative property,
Θ (|Z(t,t + h) − Z(t − h,t)| ≥ 1)
( (∣ ∣ ∣∣∣∣ Z(t−h,t)
))
∑
∣∣∣∣
= Θ Θ (Z(h,2h)(T i ) − 1)
∣ ≥ 1 Z(t − h,t)
i=1
( (
))
≤ Θ Θ |Z(h,2h)(T i ) − 1| ≥ 1 for some i ∈ {1,... ,Z(t − h,t)} | Z(t − h,t)
(
)
(2.4.4) ≤ Θ Z(t − h,t)Θ h (|Z(h,2h) − 1| ≥ 1) .
Since Z(t−h,t)Θ h (|Z(h,2h) − 1| ≥ 1) ≤ L t Θ a.s., Proposition 2.4.6 and the dominated convergence
theorem imply that the right-hand side of (2.4.4) goes to 0 as h → 0. Thus L t = L t Θ
a.s.
Likewise, there exists a random variable ̂L t with values in Z + such that, Θ a.s., Z(t−h,t+h) ↑
̂L t as h ↓ 0. Let us now notice that, for every h > 0, Θ a.s.,
Z(t − h,t + h) ≤ Z(t − h,t),
implying that ̂L t ≤ L t , Θ a.s. Moreover, thanks to Lemma 2.4.2, we have
( (v(2h) ) ) ( Z(t−h,t) (v(2h) ) )
Lt
(2.4.5) Θ(Z(t−h,t) ≥ Z(t−h,t+h)+1) = 1−Θ
≤ 1−Θ
.
v(h)
v(h)
The right-hand side of (2.4.5) tends to 0 as h → 0. So L t = ̂L t Θ a.s.
We will now establish a regularity property of the process (L t ,t ≥ 0).
Proposition 2.4.9. The process (L t ,t ≥ 0) admits a modification which is right-continuous
with left limits, and which has no fixed discontinuities.
Proof : We start the proof with three lemmas. The first one in proved in a similar but easier
way as Lemma 2.3.9.
Lemma 2.4.10. There exists λ ≥ 0 such that Θ(L t ) = e −λt for every t ≥ 0.
For every n ≥ 1 and every t ≥ 0 we set Y n
t = X 1/n
[nt] .
Lemma 2.4.11. For every t ≥ 0, Y n
t
→ L t as n → ∞, Θ a.s.
This lemma is an immediate consequence of Proposition 2.4.8.
Lemma 2.4.12. Let us define G t = σ(L s ,s ≤ t) for every t ≥ 0. Then (L t ,t ≥ 0) is a
nonnegative supermartingale with respect to the filtration (G t ,t ≥ 0).
Proof : Let s,t,s 1 ,...,s p ≥ 0 such that 0 ≤ s 1 ≤ ... ≤ s p ≤ s < t and let f : R p → R +
be a bounded measurable function. For every n ≥ 1, the offspring distribution µ 1/n is critical or
subcritical so that (X 1/n ,k ≥ 0) is a supermartingale. Thus we have
Θ
k
(
X 1/n
[nt] f (
X 1/n
[ns 1 ] ,...,X1/n [ns p]
))
≤ Θ
( (
X 1/n
[ns] f X 1/n
[ns 1 ],... ,X1/n
[ns p]
))
.
□
46

Since f is bounded and Xu
1/n ≤ L u Θ a.s. for every u ≥ 0, Lemma 2.4.11 yields
Let us set, for every t ≥ 0,
Θ ( L t f ( L s1 ,...,L sp
))
≤ Θ
(
Ls f ( L s1 ,...,L sp
))
.
˜G t = ⋂ s>tG s .
□
Recall that D denotes the set of positive dyadic numbers. From Lemma 2.4.12 and classical
results on supermartingales, we can define a right-continuous supermartingale (˜L t ,t ≥ 0) with
respect to the filtration ( ˜G t ,t ≥ 0) by setting, for every t ≥ 0,
(2.4.6) ˜Lt = lim
s↓t,s∈D L s
where the limit holds Θ a.s. and in L 1 . In a way similar to Section 2.3 we can prove that
(˜L t ,t ≥ 0) is a càdlàg modification of (L t ,t ≥ 0) with no fixed discontinuities.
From now on, to simplify notation we replace (L t ,t ≥ 0) by its càdlàg modification (˜L t ,t ≥ 0).
Proposition 2.4.13. (L t ,t ≥ 0) is a DSBP which becomes extinct Θ a.s.
Proof : By the same arguments as in the proof of (2.4.2), we can prove that, for every
0 < s < t, the following convergence holds in probability under Θ,
( )
[nt] − [ns] [nt] − [ns] + 1
(2.4.7) Z , −→
n n
L t−s.
n→∞
Let s,t,s 1 ,... ,s p ≥ 0 such that 0 ≤ s 1 ≤ ... ≤ s p ≤ s < t, λ > 0 and let f : R p −→ R be a
bounded measurable function. For every n ≥ 1, under Θ 1/n , (X 1/n
k
,k ≥ 0) is a Galton-Watson
process started at one so that
( ) ( ( ( ( ))) )
Θ 1/n f(Ys n
1
,...,Ys n
p
)exp (−λYt n ) = Θ 1/n f(Ys n
1
,... ,Ys n
p
) Θ 1/n exp −λX 1/n Y n
s
[nt]−[ns]
.
From Lemma 2.4.11, (2.4.7) and dominated convergence, we get
Θ ( f(L s1 ,...,L sp )exp(−λL t ) ) = Θ ( f(L s1 ,... ,L sp )(Θ(exp(−λL t−s ))) Ls) .
Then, (L t ,t ≥ 0) is a continuous-time Markov chain with values in Z + satisfying the branching
property. Furthermore, since H(T ) < ∞ Θ a.s., it is immediate that (L t ,t ≥ 0) becomes extinct
Θ a.s.
□
2.4.2. Identification of the probability measure Θ. Let us now define, for every T ∈ T
and t ≥ 0,
N t (T ) = #{σ ∈ T : d(ρ,σ) = t},
where we recall that ρ denotes the root of T .
Proposition 2.4.14. For every t ≥ 0, N t = L t Θ a.s.
Note that for every t ≥ 0, L t is the number of subtrees of T above level t.
Proof : Since Θ(H(T ) = 0) = 0, we have L 0 = 1 = N 0 Θ a.s. Thanks to Proposition 2.4.7,
Proposition 2.4.8 and Proposition 2.4.9, for every t > 0, Θ a.s., there exists h 0 > 0 such that for
every h ∈ (0,h 0 ],
L t = L t−h = L t+h = Z(t − h,t + h).
47

The remaining part of the argument is deterministic. We fix t,h 0 > 0 and a (deterministic)
tree T ∈ T. We assume that there is a positive integer p such that for every h ∈ (0,h 0 ],
L t = L t−h = L t+h = Z(t − h,t + h) = p,
and we will verify that N t (T ) = p. We denote by T 1 ,...,T p the p subtrees of T above level t−h 0
and we write ρ i for the root of the subtree T i . For every i ∈ {1,... ,p}, we have H(T i ) > 2h 0
so that there exists x i ∈ T i such that d(ρ i ,x i ) = 2h 0 . Let us prove that for every i ∈ {1,... ,p},
(2.4.8) T i ≤2h 0
= { σ ∈ T i : d(ρ i ,σ) ≤ 2h 0
}
= [[ρi ,x i ]].
To this end, we argue by contradiction and assume that we can find i ∈ {1,... ,p} and t i ∈ T i
such that d(ρ i ,t i ) ≤ 2h 0 and t i /∈ [[ρ i ,x i ]]. Let z i be the unique vertex of T i satisfying
[[ρ i ,z i ]] = [[ρ i ,x i ]] ∩ [[ρ i ,t i ]].
We choose c > 0 such that d(ρ i ,z i ) < c < d(ρ i ,t i ). Then it is not difficult to see that T has
at least p + 1 subtrees above level t − h 0 + c. This is a contradiction since L t−h0 +c = p. So
N t (T ) = p, which completes the proof.
□
Proposition 2.4.14 means that (L t ,t ≥ 0) is a modification of the process (N t ,t ≥ 0) which
describes the evolution of the number of individuals in the tree. Let us denote by Q the generator
of (L t ,t ≥ 0) which is of the form
⎛
Q =
⎜
⎝
⎞
0 0 0 0 0 ...
aγ(0) −a aγ(2) aγ(3) aγ(4) ...
0 2aγ(0) −2a 2aγ(2) aγ(3) ...
0 0 3aγ(0) −3a 3aγ(2) ... ⎟
⎠
. .. . .. . .. . ..
.
.
where a > 0 and γ is a critical or subcritical offspring distribution with γ(1) = 0.
For every t ≥ 0 we let F t be the σ-field on T generated by the mapping T ↦−→ T ≤t and
completed with respect to Θ. Thus (F t ,t ≥ 0) is a filtration on T.
Lemma 2.4.15. Let t > 0 and p ∈ N. Under Θ, conditionally on F t and given {L t = p}, the
p subtrees of T above level t are independent and distributed according to Θ.
Proof : Thanks to Lemma 2.2.2 and Lemma 2.4.3, we can construct on the same probability
space (Ω,P), a sequence of T-valued random variables (T n ) n≥1 distributed according to Θ 1/n
and a sequence of A-valued random variables (θ n ) n≥1 distributed according to Π µ1/n such that,
for every n ≥ 1,
)
(2.4.9) GH
(T n ,n −1 T θn ≤ 4n −1 .
For every n ≥ 1 and k ≥ 0, we define X n k = #{u ∈ θ n : |u| = k}. Let t ≥ 0 and p ≥ 1, let
g : T −→ R be a bounded continuous function and let G : T p −→ R be a bounded continuous
symmetric function. For n ≥ 1, on the event {X n [nt]
= p}, we set {u n 1 ,... ,un p } = {u ∈ θ n :
|u| = [nt]} and θ i n = τ u n
i
θ n for every i ∈ {1,... ,p}. Then we can write, thanks to the branching
property of Galton-Watson trees,
( ) (
)
E g n
(½nX )
−1 T θn
n
[nt] =po ≤[nt]
G n −1 T θ1 n,... ,n −1 T θp n
( ) ) ) ⊗p (
))
(2.4.10) = g n E(½nX −1 T θn
n
[nt] =po ≤[nt]
(Π µ1/n G
(n −1 T θ 1
,... ,n −1 T θp ,
,
48

where θ 1 ,... ,θ p denote the coordinate variables under the product measure (Π µ1/n ) ⊗p . As a
consequence of (2.4.9), we see that the law of n −1 T θ under Π µ1/n converges to Θ in the sense of
weak convergence of measures on the space T. Then, thanks to Lemma 2.4.11, the right-hand
side of (2.4.10) converges as n → ∞ to
Θ (½{L t=p}g(T ≤t ) ) Θ ⊗p (G(T 1 ,... , T p )).
Similarly, the left-hand side of (2.4.10) converges as n → ∞ to
Θ (½{L t=p}g(T ≤t )G(T 1 ,... , T p ) ) ,
where T 1 ,...,T p are the p subtrees of T above level t on the event {L t = p}. This completes
the proof.
□
Let us define J = inf{t ≥ 0 : L t ≠ 1}. Then J is an (F t ) t≥0 -stopping time.
Lemma 2.4.16. Let p ∈ N. Under Θ, given {L J = p}, the p subtrees of T above level J are
independent and distributed according to Θ, and are independent of J.
Proof : Let p ∈ N, let f : R + −→ R be a bounded continuous function and let G : T p −→ R
be a bounded continuous symmetric function. On the event {L J = p}, we denote by T 1 ,...,T p
the p subtrees of T above level J. Let n ≥ 1 and k ≥ 0. On the event {L (k+1)/n = p}, we
denote by T 1,(n,k) ,... , T p,(n,k) the p subtrees of T above level (k + 1)/n. On the one hand, the
right-continuity of the mapping t ↦−→ L t gives
Θ
( ∞
∑
k=1½{L (k+1)/n =p} G (
T 1,(n,k) ,... , T p,(n,k)) f ((k + 1)/n)½{k/n

CHAPITRE 3
Conditioned Brownian trees
3.1. Introduction
In this work, we define and study a continuous tree of one-dimensional Brownian paths
started from the origin, which is conditioned to remain in the positive half-line. An important
motivation for introducing this object comes from its relation with analogous discrete models
which are discussed in several recent papers.
In order to present our main results, let us briefly describe a construction of unconditioned
Brownian trees. We start from a positive Brownian excursion conditioned to have duration 1 (a
normalized Brownian excursion in short), which is denoted by (e(s),0 ≤ s ≤ 1). This random
function can be viewed as coding a continuous tree via the following simple prescriptions. For
every s,s ′ ∈ [0,1], we set
m e (s,s ′ ) := inf e(r).
s∧s ′ ≤r≤s∨s ′
We then define an equivalence relation on [0,1] by setting s ∼ s ′ if and only if e(s) = e(s ′ ) =
m e (s,s ′ ). Finally we put
d e (s,s ′ ) = e(s) + e(s ′ ) − 2m e (s,s ′ )
and note that d e (s,s ′ ) only depends on the equivalence classes of s and s ′ . Then the quotient
space T e := [0,1]/ ∼ equipped with the metric d e is a compact R-tree (see e.g. Section 2 of
[22]). In other words, it is a compact metric space such that for any two points σ and σ ′ there is
a unique arc with endpoints σ and σ ′ and furthermore this arc is isometric to a compact interval
of the real line. We view T e as a rooted R-tree, whose root ρ is the equivalence class of 0. For
every σ ∈ T e , the ancestral line of σ is the line segment joining ρ to σ. This line segment is
denoted by [[ρ,σ]]. We write ṡ for the equivalence class of s, which is a vertex in T e at generation
e(s) = d e (0,s).
Up to unimportant scaling constants, T e is the Continuum Random Tree (CRT) introduced
by Aldous [3]. The preceding presentation is indeed a reformulation of Corollary 22 in [5], which
was proved via a discrete approximation (a more direct approach was given in [35]). As Aldous
[5] has shown, the CRT is the scaling limit of critical Galton-Watson trees conditioned to have
a large fixed progeny (see [20] and [22] for recent generalizations of Aldous’ result). The fact
that Brownian excursions can be used to model continuous genealogies had been used before, in
particular in the Brownian snake approach to superprocesses (see [34]).
We can now combine the branching structure of the CRT with independent spatial motions.
We restrict ourselves to spatial displacements given by linear Brownian motions, which is the
case of interest in this work. Conditionally given e, we introduce a centered Gaussian process
(V σ ,σ ∈ T e ) with covariance
cov(Vṡ,Vṡ′) = m e (s,s ′ ) , s,s ′ ∈ [0,1].
51

This definition should become clear if we observe that m e (s,s ′ ) is the generation of the most
recent common ancestor to ṡ and ṡ ′ in the tree T e . It is easy to verify that the process (V σ ,σ ∈
T e ) has a continuous modification. The random measure Z on R defined by
〈Z,ϕ〉 =
∫ 1
0
ϕ(Vṡ)ds
is then the one-dimensional Integrated Super-Brownian Excursion (ISE, see Aldous [6]). Note
that ISE in higher dimensions, and related Brownian trees, have appeared recently in various
asymptotic results for statistical mechanics models (see e.g. [18],[28],[29]). The support, or
range, of ISE is
R := {V σ : σ ∈ T e }.
For our purposes, it is also convenient to reinterpret the preceding notions in terms of the
Brownian snake. The Brownian snake (W s ,0 ≤ s ≤ 1) driven by the normalized excursion e is
obtained as follows (see subsection 3.2.1 for a more detailed presentation). For every s ∈ [0,1],
W s = (W s (t),0 ≤ t ≤ e(s)) is the finite path which gives the spatial positions along the ancestral
line of ṡ : W s (t) = V σ if σ is the vertex at distance t from the root on the segment [[ρ,ṡ]]. Note
that W s only depends on the equivalent class ṡ. We view W s as a random element of the space
W of finite paths.
Our first goal is to give a precise definition of the Brownian tree (V σ ,σ ∈ T e ) conditioned to
remain positive. Equivalently this amounts to conditioning ISE to put no mass on the negative
half-line. Our first theorem gives a precise meaning to this conditioning in terms of the Brownian
snake. We denote by N (1)
0 the distribution of (W s ,0 ≤ s ≤ 1) on the canonical space C([0,1], W)
of continuous functions from [0,1] into W, and we abuse notation by still writing (W s ,0 ≤ s ≤ 1)
for the canonical process on this space. The range R is then defined under N (1)
0 by
}
R =
{Ŵs : 0 ≤ s ≤ 1
where Ŵs denotes the endpoint of the path W s .
Theorem 3.1.1. We have
lim ε −4 N (1)
2
0
(R ⊂] − ε, ∞[) =
ε↓0 21 .
There exists a probability measure on C([0,1], W), which is denoted by N (1)
0 , such that
lim N (1)
0 ε↓0
(· | R ⊂] − ε, ∞[) = N(1) 0 ,
in the sense of weak convergence in the space of probability measures on C([0,1], W).
Our second theorem gives an explicit representation of the conditioned measures N (1)
0 , which
is analogous to a famous theorem of Vervaat [52] relating the normalized Brownian excursion
to the Brownian bridge. To state this result, we need the notion of re-rooting. For s ∈ [0,1], we
write T [s]
e for the “same” tree T e but with root ṡ instead of ρ = ˙0. We then shift the spatial
positions by setting V σ
[s] = V σ − Vṡ for every σ ∈ T e , in such a way that the spatial position of
the new root is still the origin. (Notice that both T e
[s] and V [s] only depend on ṡ, and we could
as well define T [σ]
e and V [σ] for σ ∈ T e .) Finally, the re-rooted snake W [s] = (W r [s] ,0 ≤ r ≤ 1) is
defined analogously as before : For every r ∈ [0,1], W r
[s] is the path giving the spatial positions
V σ [s] along the ancestral line (in the re-rooted tree) of the vertex s + r mod. 1.
52

Theorem 3.1.2. Let s ∗ be the unique time of the minimum of Ŵ on [0,1]. The probability
measure N (1)
0 is the law under N (1)
0 of the re-rooted snake W [s∗] .
If we want to define one-dimensional ISE conditioned to put no mass on the negative half-line,
the most natural way is to condition to put no mass on ] − ∞, −ε[ and then to let ε go to 0.
As a consequence of the previous two theorems, this is equivalent to shifting the unconditioned
ISE to the right, so that the left-most point of its support becomes the origin. Another method
would be to condition the mass in ]−∞,0] to be less than ε and then to let ε go to 0. Proposition
3.3.7 below shows that this leads to the same measure N (1)
0 .
Both Theorem 3.1.1 and Theorem 3.1.2 could be presented in a different and perhaps more
elegant manner by using the formalism of spatial trees as in Section 5 of [22]. In this formalism,
a spatial tree is a pair (T,U) where T is a compact rooted R-tree (in fact an equivalent class
of such objects modulo root-preserving isometries) and U is a continuous mapping from T into
R d . Then the second assertion of Theorem 3.1.1 can be rephrased by saying that the conditional
distribution of the spatial tree (T e ,V ) knowing that R ⊂] − ε, ∞[ has a limit when ε goes to 0,
and Theorem 3.1.2 says that this limit is the distribution of (T [σ∗]
e ,V [σ∗] ) where σ ∗ is the unique
vertex minimizing V . We have chosen the above presentation because the Brownian snake plays
a fundamental role in our proofs and also because the resulting statements are stronger than
the ones in terms of spatial trees.
Let us discuss the relationship of the above theorems with previous results. The first assertion
of Theorem 3.1.1 is closely related to some estimates of Abraham and Werner [1]. In particular,
Abraham and Werner proved that the probability for a Brownian snake driven by a Brownian
excursion of height 1 not to hit the set ]−∞, −ε[ behaves like a constant times ε 4 (see Section 3.4
below). The d-dimensional Brownian snake conditioned not to exit a domain D was studied by
Abraham and Serlet [2], who observed that this conditioning gives rise to a particular instance
of the Brownian snake with drift. The setting in [2] is different from the present work, in that
the initial point of the snake lies inside the domain, and not at its boundary as here. We also
mention the paper [32] by Jansons and Rogers, who establish a decomposition at the minimum
for a Brownian tree where branchings occur only at discrete times.
An important motivation for the present work came from several recent papers that discuss
asymptotics for planar maps. Cori and Vauquelin [15] proved that there exists a bijection between
rooted planar quadrangulations and certain discrete trees called well-labelled trees (see
also Chassaing & Schaeffer [14] for a more tractable description of this bijection). Roughly, a
well-labelled tree consists of a (discrete) plane tree whose vertices are given labels which are
positive integers, with the constraints that the label of the root is 1 and the labels of two neighboring
vertices can differ by at most 1. Our conditioned Brownian snake should then be viewed
as a continuous model for well-labelled trees. This idea was exploited in [14] and especially in
Marckert and Mokkadem [45], where the re-rooted snake W [s∗] appears in the description of the
Brownian map, which is the continuous object describing scaling limits of planar quadrangulations.
In contrast with the present work, the re-rooted snake W [s∗] is not interpreted in [45] as
a conditioned object, but rather as a scaling limit of re-rooted discrete snakes. Closely related
models of discrete labelled trees are also of interest in theoretical physics : See in particular [9]
and [10]. The article [39], which was motivated by [14] and [45], proves that our conditioned
Brownian tree is the scaling limit of discrete spatial trees conditioned to remain positive. To be
specific, consider a Galton-Watson tree whose offspring distribution is critical and has (small)
exponential moments, and condition this tree to have exactly n vertices (in the special case of
53

the geometric distribution, this gives rise to a tree that is uniformly distributed over the set of
plane trees with n vertices). This branching structure is combined with a spatial displacement
which is a symmetric random walk with bounded jump size on Z. Assuming that the root is at
the origin of Z, the spatial tree is then conditioned to remain on the positive side. According to
the main theorem of [39], the scaling limit of this conditioned discrete tree when n → ∞ leads
to the measure N (1)
0 discussed above. The convergence here, and the precise form of the scaling
transformation, are as in Theorem 2 of [31], which discusses scaling limits for unconditioned
discrete snakes.
Let us now describe the other contributions of this paper. Although the preceding theorems
have been stated for the measure N (1)
0 , a more fundamental object is the excursion measure N 0 of
the Brownian snake (see e.g. [37]). Roughly speaking, N 0 is obtained by the same construction as
above, but instead of considering a normalized Brownian excursion, we now let e be distributed
according to the (infinite) Itô measure of Brownian excursions. If σ(e) denotes the duration
of excursion e, we have N (1)
0 = N 0 (· | σ = 1). It turns out that many calculations are more
tractable under the infinite measure N 0 than under N (1)
0 . For this reason, both Theorems 3.1.1
and Theorem 3.1.2 are proved in Section 3.3 as consequences of Theorem 3.3.1, which deals
with N 0 . Motivated by Theorem 3.3.1 we introduce another infinite measure denoted by N 0 ,
which should be interpreted as N 0 conditioned on the event {R ⊂ [0, ∞[}, even though the
conditioning requires some care as we are dealing with infinite measures. In the same way as
for unconditioned measures, we have N (1)
0 = N 0 (· | σ = 1). Another motivation for considering
the measure N 0 comes from connections with superprocesses : Analogously to Chapter IV of
[37] in the unconditioned case, N 0 could be used to define and to analyse a one-dimensional
super-Brownian motion started from the Dirac measure δ 0 and conditioned never to charge the
negative half-line.
In Section 3.4, we present a different approach that leads to the same limiting measures. If
H(e) stands for the height of excursion e, we consider for every h > 0 the measure N h 0 := N 0(· |
H = h). In the above construction this amounts to replacing the normalized excursion e by a
Brownian excursion with height h. By using a famous decomposition theorem of Williams, we
can then analyse the behavior of the measure N h 0 conditioned on the event that the range does
not intersect ] − ∞, −ε[ and show that it has a limit denoted by N h 0 when ε → 0. The method
also provides information about the Brownian tree under N h 0 : This Brownian tree consists of a
spine whose distribution is absolutely continuous with respect to that of the nine-dimensional
Bessel process, and as usual a Poisson collection of subtrees originating from the spine, which
are Brownian snake excursions conditioned not to hit the negative half-line. The connection with
the measures N (1)
0 and N 0 is made by proving that N h 0 = N 0(· | H = h). Several arguments in
this section have been inspired by Abraham and Werner’s paper [1]. It should also be noted that
a discrete version of the nine-dimensional Bessel process already appears in the paper [13] by
Chassaing and Durhuus.
At the end of Section 3.4, we also discuss the limiting behavior of the measures N h 0 as h → ∞.
This leads to a probability measure N ∞ 0 that should be viewed as the law of an infinite Brownian
snake excursion conditioned to stay positive. We again get a description of the Brownian tree
coded by N ∞ 0 in terms of a spine and conditioned Brownian snake excursions originating from
this spine. Moreover, the description is simpler in the sense that the spine is exactly distributed
as a nine-dimensional Bessel process started at the origin.
54

Section 3.5 gives an explicit formula for the finite-dimensional marginal distributions of the
Brownian tree under N 0 , that is for
)
N 0
( ∫
]0,σ[ p ds 1 ...ds p F(W s1 ,...,W sp )
where p ≥ 1 is an integer and F is a symmetric nonnegative measurable function on W p . In
a way similar to the corresponding result for the unconditioned Brownian snake (see (3.2.1)
below), this formula involves combining the branching structure of certain discrete trees with
spatial displacements. Here however because of the conditioning, the spatial displacements turn
out to be given by nine-dimensional Bessel processes rather than linear Brownian motions. In
the same way as the finite-dimensional marginal distributions of the CRT can be derived from
the analogous formula under the Itô measure (see Chapter III of [37]), one might hope to derive
the expression of the finite-dimensional marginals under N (1)
0 from the case of N 0 . This idea
apparently leads to untractable calculations, but we still expect Theorem 3.5.1 to have useful
applications in future work about conditioned trees.
Basic facts about the Brownian snake are recalled in Section 3.2, which also establishes a few
important preliminary results, some of which are of independent interest. In particular, we state
and prove a general version of the invariance property of N 0 under re-rooting (Theorem 3.2.3).
This result is clearly related to the invariance of the CRT under uniform re-rooting, which was
observed by Aldous [4] (and generalized to Lévy trees in Proposition 4.8 of [22]). An equivalent
form of Theorem 3.2.3 already appears as Proposition 4.9 of [45] : See the discussion after the
statement of this theorem in subsection 3.2.3.
3.2. Preliminaries
In this section, we recall the basic facts about the Brownian snake that we will use later,
and we also establish a few important preliminary results. We refer to [37] for a more detailed
presentation of the Brownian snake and its connections with partial differential equations. In
the first four subsections below, we deal with the d-dimensional Brownian snake since the proofs
are not more difficult in that case, and the results may have other applications.
3.2.1. The Brownian snake. The (d-dimensional) Brownian snake is a Markov process
taking values in the space W of finite paths in R d . Here a finite path is simply a continuous
mapping w : [0,ζ] −→ R d , where ζ = ζ (w) is a nonnegative real number called the lifetime of w.
The set W is a Polish space when equipped with the distance
d(w,w ′ ) = |ζ (w) − ζ (w ′ )| + sup |w(t ∧ ζ (w) ) − w ′ (t ∧ ζ (w ′ ))|.
t≥0
The endpoint (or tip) of the path w is denoted by ŵ. The range of w is denoted by w[0,ζ (w) ].
In this work, it will be convenient to use the canonical space Ω := C(R + , W) of continuous
functions from R + into W, which is equipped with the topology of uniform convergence on every
compact subset of R + . The canonical process on Ω is then denoted by
and we write ζ s = ζ (Ws) for the lifetime of W s .
W s (ω) = ω(s) , ω ∈ Ω ,
Let w ∈ W. The law of the Brownian snake started from w is the probability measure P w
on Ω which can be characterized as follows. First, the process (ζ s ) s≥0 is under P w a reflected
55

Brownian motion in [0, ∞[ started from ζ (w) . Secondly, the conditional distribution of (W s ) s≥0
knowing (ζ s ) s≥0 , which is denoted by Θ ζ w, is characterized by the following properties :
(i) W 0 = w, Θ ζ w a.s.
(ii) The process (W s ) s≥0 is time-inhomogeneous Markov under Θ ζ w. Moreover, if 0 ≤ s ≤ s ′ ,
• W s ′(t) = W s (t) for every t ≤ m(s,s ′ ) := inf [s,s ′ ] ζ r , Θ ζ w a.s.
• (W s ′(m(s,s ′ ) + t) − W s ′(m(s,s ′ ))) 0≤t≤ζs ′ −m(s,s ′ ) is independent of W s and distributed
as a d-dimensional Brownian motion started at 0 under Θ ζ w.
Informally, the value W s of the Brownian snake at time s is a random path with a random
lifetime ζ s evolving like reflecting Brownian motion in [0, ∞[. When ζ s decreases, the path is
erased from its tip, and when ζ s increases, the path is extended by adding “little pieces” of
Brownian paths at its tip.
Excursion measures play a fundamental role throughout this work. We denote by n(de) the
Itô measure of positive Brownian excursions. This is a σ-finite measure on the space C(R + , R + )
of continuous functions from R + into R + . We write
σ(e) = inf{s > 0 : e(s) = 0}
for the duration of excursion e. For s > 0, n (s) will denote the conditioned measure n(· | σ = s).
Our normalization of the excursion measure is fixed by the relation
∫ ∞
ds
n =
2 √ 2πs n (s).
3
0
If x ∈ R d , the excursion measure N x of the Brownian snake from x is then defined by
∫
N x = n(de) Θx
e
C(R + ,R + )
where x denotes the trivial element of W with lifetime 0 and initial point x. Alternatively, we
can view N x as the excursion measure of the Brownian snake from the regular point x. With a
slight abuse of notation we will also write σ(ω) = inf{s > 0 : ζ s (ω) = 0} for ω ∈ Ω. We can then
consider the conditioned measures
∫
N x (s) = N x(· | σ = s) = n (s) (de) Θx e .
C(R + ,R + )
Note that in contrast to the introduction we now view N (s)
x as a measure on Ω rather than on
C([0,s], W). The range R = R(ω) is defined by R = {Ŵs : s ≥ 0}.
Lemma 3.2.1. Suppose that d = 1 and let x > 0.
(i) We have
N x (R∩] − ∞,0] ≠ ∅) = 3
2x 2.
(ii) For every λ > 0,
√
N x
(1 −½{R∩]−∞,0]=∅} e −λσ) λ
(
)
= 3(coth(2 1/4 xλ 1/4 )) 2 − 2
2
where coth(y) = cosh(y)/sinh(y).
Proof : (i) According to Section VI.1 of [37], the function u(x) = N x (R∩] − ∞,0] ≠ ∅)
solves u ′′ = 4u 2 in ]0, ∞[, with boundary condition u(0+) = +∞. The desired result follows.
(ii) See Lemma 7 in [17].
56
□

3.2.2. Finite-dimensional marginal distributions. In this subsection we state a result
giving information about the joint distribution of the values of the Brownian snake at a finite
number of times and its range. In order to state this result, we need some formalism for trees.
We first introduce the set of labels
∞⋃
U = {1,2} n
n=0
where by convention {1,2} 0 = {∅}. An element of U is thus a sequence u = u 1 ...u n of elements
of {1,2}, and we set |u| = n, so that |u| represents the “generation” of u. In particular, |∅| = 0.
The mapping π : U\{∅} −→ U is defined by π(u 1 ... u n ) = u 1 ...u n−1 (π(u) is the “father” of
u). In particular, if k = |u|, we have π k (u) = ∅.
A binary (plane) tree T is a finite subset of U such that :
(i) ∅ ∈ T .
(ii) u ∈ T \{∅} ⇒ π(u) ∈ T .
(iii) For every u ∈ T , either u1 ∈ T and u2 ∈ T , or u1 /∈ T and u2 /∈ T (u is called a leaf
in the second case).
We denote by A the set of all binary trees. A marked tree is then a pair (T ,(h u ) u∈T ) where
T ∈ A and h u ≥ 0 for every u ∈ T . We denote by T the space of all marked trees. In this
work it will be convenient to view marked trees as R-trees in the sense of [22] or [24] (see
also Section 3.1 above). This can be achieved through the following explicit construction. Let
θ = (T ,(h u ) u∈T ) be a marked tree and let R T be the vector space of all mappings from T into
R. Write (ε u ,u ∈ T ) for the canonical basis of R T . Then consider the mapping
p θ : ⋃
{u} × [0,h u ] −→ R T
defined by
u∈T
|u|
∑
p θ (u,l) = h π k (u) ε π k (u) + lε u .
k=1
As a set, the R-tree associated with θ is the range ˜θ of p θ . Note that this is a connected union
of line segments in R T . It is equipped with the distance d θ such that d θ (a,b) is the length of the
shortest path in ˜θ going from a to b. By definition, the range of this path is the segment between
a and b and is denoted by [[a,b]]. Finally, we will write L θ for the (one-dimensional) Lebesgue
measure on ˜θ.
By definition, leaves of ˜θ are points of the form p θ (u,h u ) where u is leaf of θ. Points of the
form p θ (u,h u ) when u is not a leaf are called nodes of ˜θ. We write L(θ) for the set of leaves of
˜θ, and I(θ) for the set of its nodes. The root of ˜θ is just the point 0 = p θ (∅,0).
We will consider Brownian motion indexed by ˜θ, with initial point x ∈ R d . Formally, we may
consider, under the probability measure Q θ x , a collection (ξu ) u∈T of independent d-dimensional
Brownian motions all started at 0 except ξ ∅ which starts at x, and define a continuous process
(V a ,a ∈ ˜θ) by setting
|u|
∑ ( )
V pθ (u,l) = ξ πk (u)
h π k (u) + ξ u (l),
k=1
57

for every u ∈ T and l ∈ [0,h u ]. Finally, with every leaf a of ˜θ we associate a stopped path w (a)
with lifetime d θ (0,a) : For every t ∈ [0,d θ (0,a)], w (a) (t) = V r(a,t) where r(a,t) is the unique
element of [[0,a]] such that d θ (0,r(a,t)) = t.
For every integer p ≥ 1, denote by A p the set of all binary trees with p leaves, and by T p the
corresponding set of marked trees. The uniform measure Λ p on T p is defined by
∫
Λ p (dθ)F(θ) = ∑ ∫ ∏
dh v F(T ,(h v ) v∈T ).
T p T ∈A p v∈T
With this notation, Proposition IV.2 of [37] states that, for every integer p ≥ 1 and every
symmetric nonnegative measurable function F on W p ,
( ∫ ) ∫ [ ( (
(3.2.1) N x ds 1 ...ds p F(W s1 ,... ,W sp ) = 2 p−1 p! Λ p (dθ)Q θ x F w (a)) .
]0,σ[ a∈L(θ))]
p
We will need a stronger result concerning the case where the function F also depends on the
range R of the Brownian snake. To state this result, denote by K the space of all compact subsets
of R d , which is equipped with the Hausdorff metric and the associated Borel σ-field. Suppose
that under the probability measure Q θ x (for each choice of θ in T), in addition to the process
(V a ,a ∈ ˜θ), we are also given an independent Poisson point measure on ˜θ × Ω, denoted by
∑
δ (ai ,ω i ),
with intensity 4 L θ (da) ⊗ N 0 (dω).
i∈I
Theorem 3.2.2. For every nonnegative measurable function F on W p × K × R + , which is
symmetric in the first p variables, we have
)
N x
( ∫
]0,σ[ p ds 1 ...ds p F(W s1 ,...,W sp , R,σ)
∫
= 2 p−1 p!
Λ p (dθ)Q θ x
where cl(A) denotes the closure of the set A.
[
F
( (
w (a)) a∈L(θ) ,cl ( ⋃
i∈I
(V ai + R(ω i ))
)
, ∑ i∈I
σ(ω i )
Remark. It is immediate to see that
( ) ⎛
⋃
cl (V ai + R(ω i )) = ⎝ ⋃ [ w (a) 0,ζ (w )] ⎞ ( )
⋃
⎠ (a) ∪ (V ai + R(ω i )) , Q θ x a.e.
i∈I
i∈I
a∈L(θ)
Proof : Consider first the case p = 1. Let F 1 be a nonnegative measurable function on W,
and let F 2 and F 3 be two nonnegative measurable functions on Ω. By applying the Markov
property under N x at time s, then using the time-reversal invariance of N x (which is easy from
the analogous property for the Itô measure n(de)), and finally using the Markov property at
)]
,
58

time s once again, we get
(∫ σ ( ) ( ) )
N x ds F 1 (W s )F 2 (W (s−r) +) r≥0 F 3 (W s+r ) r≥0
0
= N x
(∫ σ
0
= N x
(∫ σ
0
= N x
(∫ σ
0
ds F 1 (W s )F 2
(
(W (s−r) +) r≥0
)
E Ws
[
F 3
(
(W r∧σ ) r≥0
)] )
ds F 1 (W s )F 2
(
(W s+r ) r≥0
)
E Ws
[
F 3
(
(W r∧σ ) r≥0
)] )
ds F 1 (W s ) E Ws
[
F 2
(
(W r∧σ ) r≥0
)]
E Ws
[
F 3
((W r∧σ ) r≥0
)] ) .
We then use the case p = 1 of (3.2.1) to see that the last quantity is equal to
∫ ∞
∫
[ ( )] [ (
dt Px t (dw)F 1(w) E w F 2 (W r∧σ ) r≥0 E w F 3 (W r∧σ ) r≥0
)],
0
where Px t denotes the law of Brownian motion started at x and stopped at time t (this law is
viewed as a probability measure on W). Now if we specialize to the case where F 2 is a function
of the form F 2 (ω) = G 2 ({Ŵs(ω) : s ≥ 0},σ), an immediate application of Lemma V.2 in [37]
shows that
⎡ ⎛ ⎛
⎞ ⎞⎤
[ ( )]
E w F 2 (W r∧σ ) r≥0 = E ⎣G 2
⎝cl (w(t j ) + R(ω j )) ⎠ , ∑ σ(ω j ) ⎠⎦ ,
j∈J
⎝ ⋃ j∈J
where ∑ j∈J δ (t j ,ω j ) is a Poisson point measure on [0,ζ (w) ]×Ω with intensity 2 dt N 0 (dω). Applying
the same observation to F 3 , we easily get the case p = 1 of the theorem.
The general case can be derived along similar lines by using Theorem 3 in [35]. Roughly
speaking, the case p = 1 amounts to combining Bismut’s decomposition of the Brownian excursion
(Lemma 1 in [35]) with the spatial displacements of the Brownian snake. For general p, the
second assertion of Theorem 3 in [35] provides the analogue of Bismut’s decomposition, which
when combined with spatial displacements leads to the statement of Theorem 3.2.2. Details are
left to the reader.
□
3.2.3. The re-rooting theorem. In this subsection, we state and prove an important
invariance property of the Brownian snake under N 0 , which plays a major role in Section 3.3
below. We first need to introduce some notation. For every s,r ∈ [0,σ], we set
{
s + r if s + r ≤ σ ,
s ⊕ r =
s + r − σ if s + r > σ .
We also use the following convenient notation for closed intervals : if u,v ∈ R, [u,v] = [v,u] =
[u ∧ v,u ∨ v].
Let s ∈ [0,σ[. In order to define the re-rooted snake W [s] , we first set
if r ∈ [0,σ], and ζ r
[s]
that
ζ [s]
r = ζ s + ζ s⊕r − 2 inf
u∈[s,s⊕r] ζ u ,
= 0 if r > σ. We also want to define the stopped paths W [s]
r , in such a way
Ŵ [s]
r = Ŵs⊕r − Ŵs ,
59

if r ∈ [0,σ], and Ŵ r
[s] = 0 if r > σ. To this end, we may notice that Ŵ [s] satisfies the property
Ŵ [s]
r
= Ŵ [s]
r ′
if ζ [s]
r
= ζ [s]
r ′
= inf
u∈[r,r ′ ] ζ[s] u
and so in the terminology of [44], (W r [s] ) 0≤r≤σ is uniquely determined as the snake whose tour
is (ζ r [s] ,Ŵ r [s] ) 0≤r≤σ (see the homeomorphism theorem of [44]). We have the explicit formula, for
r ≥ 0, and 0 ≤ t ≤ ζ r [s] ,
(3.2.2) W [s]
r (t) = Ŵ [s]
sup{u≤r : ζ [s]
u =t} .
As explained in the introduction, (ζ r [s] ) r≥0 codes the same R-tree as the one coded by (ζ r ) r≥0 ,
but with a new root which is the vertex originally labelled by s, and Ŵ r
[s] gives the spatial
displacements along the line segment from the (new) root to the vertex coded by r (in the
coding given by ζ [s] ).
Theorem 3.2.3. For every nonnegative measurable function F on R + × Ω,
(∫ σ
N 0 ds F
(s,W [s])) (∫ σ
)
= N 0 ds F(s,W) .
0
Remark. For every s ∈ [0,σ[, the duration of the re-rooted snake excursion W [s] is the same
as that of the original one. Using this simple observation, and replacing F by½{1−ε 0). Via
a continuity argument, it follows that, for every s ∈ [0,1[, and every nonnegative measurable
function G on Ω,
(3.2.3) N (1)
0
( (
G W [s])) = N (1)
0 (G(W)).
The identity (3.2.3) appears as Proposition 4.9 of [45], which is proved via discrete approximations.
Note that conversely, it would be easy to derive Theorem 3.2.3 from (3.2.3). We have
chosen to give an independent proof of Theorem 3.2.3 because this result plays a major role
in the present work, and also because the proof below fits in better with our general strategy,
which is to deal first with unnormalized excursion measures before conditioning with respect to
the duration.
Proof : By (3.2.2), W [s] can be written N 0 a.e. as Φ(ζ [s] ,Ŵ [s] ), where the deterministic
function Φ does not depend on s. Also note that when s = 0, W = W [0] = Φ(ζ,Ŵ), N 0 a.e. In
view of these considerations, it will be sufficient to treat the case when
)
F(s,W) = F 1 (s,ζ)F 2
(s,Ŵ
where F 1 and F 2 are nonnegative measurable functions defined respectively on R + ×C(R + , R + )
and on R + × C(R + , R d ). We first deal with the special case F 2 = 1.
For s ∈ [0,σ[ and r ≥ 0, set
ζ 1,s
r = ζ (s−r) + − ζ s ,
ζ 2,s
r = ζ s+r − ζ s .
0
60

Let G be a nonnegative measurable function on R + × C(R + , R) × R + × C(R + , R). From the
Bismut decomposition of the Brownian excursion (see e.g. Lemma 1 in [35]), we have
(∫ σ (
) ) ∫ ∞ [ (
)]
N 0 ds G s,(ζr 1,s ) r≥0 ,σ − s,(ζr 2,s ) r≥0 = daE G T a ,(B r∧Ta ) r≥0 ,T a ′ ,(B′ r∧T a ′) r≥0 ,
0
where B and B ′ are two independent linear Brownian motions started at 0, and
Now observe that
T a = inf{r ≥ 0 : B r = −a},
T ′ a = inf{r ≥ 0 : B ′ r = −a}.
0
ζ r [s] = ζr
2,s − 2 inf
0≤u≤r ζ2,s u , if 0 ≤ r ≤ σ − s ,
ζ [s]
σ−r = ζ1,s r − 2 inf
0≤u≤r ζ1,s u , if 0 ≤ r ≤ s ,
and note that R t := B t − 2inf r≤t B r and R r ′ := B′ t − 2inf r≤t B r ′ are two independent threedimensional
Bessel processes, for which
L a := sup{t ≥ 0 : R t ≤ a} = T a ,
L ′ a := sup{t ≥ 0 : R ′ t ≤ a} = T ′ a .
(This is Pitman’s theorem, see e.g. [50], Theorem VI.3.5.) It follows that
(∫ σ (
) )
N 0 ds G σ − s,(ζ [s]
r∧(σ−s) ) r≥0,s,(ζ [s]
σ−(r∧s) ) r≥0
0
∫ ∞ [
))
= daE G
(L ′ a,(R r∧L ′ ′ ) r≥0,L
a a ,(R r∧La ) r≥0
0
(∫ σ
)
= N 0 ds G
(s,(ζ )
r∧s ) r≥0 ,σ − s,(ζ (σ−r)∨s ) r≥0
0
where the last equality is again a consequence of the Bismut decomposition, together with the
Williams reversal theorem ([50], Corollary XII.4.4). Changing s into σ − s in the last integral
gives the desired result when F 2 = 1.
Let us consider the general case. For simplicity we take d = 1, but the argument can obviously
be extended. From the definition of the Brownian snake, we have
(∫ σ
)
N 0 ds F 1 (s,ζ)F 2
(s,Ŵ
) ( ∫ σ
( )])
= N 0 ds F 1 (s,ζ)Θ ζ 0
[F 2 s,Ŵ
0
and Ŵ is under Θζ 0 a centered Gaussian process with covariance
)
cov Θ
ζ
(Ŵs ,Ŵs ′ = inf ζ r.
0
r∈[s,s ′ ]
We have in particular
(∫ σ (
N 0 ds F 1 s,ζ [s]) (
F 2 s,Ŵ [s]))
0
(∫ σ
= N 0 ds F 1
(s,ζ [s]) ( )
Θ ζ 0
[F 2 s,(Ŵs⊕r − Ŵs
0
0
r≥0
)])
.
Now note that (Ŵs⊕r − Ŵs) r≥0 is under Θ ζ 0 a Gaussian process with covariance
)
cov
(Ŵs⊕r − Ŵs,Ŵs⊕r ′ − Ŵs = inf ζ
[s⊕r,s⊕r ′ u − inf ζ u − inf ζ
] [s⊕r,s] [s⊕r ′ u + ζ s = inf
,s]
61
[r,r ′ ] ζ[s] u ,

where the last equality follows from an elementary verification. Hence,
( ) )] ( )]
Θ ζ 0
[F 2 s,(Ŵs⊕r − Ŵs = Θ ζ[s]
0
[F 2 s,Ŵ ,
and, using the first part of the proof,
(∫ σ
N 0
0
This completes the proof.
r≥0
ds F 1
(
s,ζ [s]) F 2
(
s,Ŵ [s])) = N 0
(∫ σ
0
(
ds F 1 s,ζ [s]) ( )]
Θ ζ[s]
0
[F )
2 s,Ŵ
(∫ σ
= N 0 ds F 1 (s,ζ)Θ ζ 0
0
( )] [F )
2 s,Ŵ
(∫ σ (s,Ŵ) )
= N 0 ds F 1 (s,ζ)F 2 .
0
□
3.2.4. The special Markov property. Let D be a domain in R d , and fix a point x ∈ D.
For every w ∈ W, we set
τ(w) := inf{t ≥ 0 : w(t) /∈ D}
where inf ∅ = +∞ as usual. The random set
{s ≥ 0 : τ(W s ) < ζ s }
is open N x a.e., and can thus be written as a disjoint union of open intervals ]a i ,b i [, i ∈ I. It is
easy to verify that N x a.e. for every i ∈ I and every s ∈]a i ,b i [,
τ (W s ) = τ (W ai ) = τ (W bi ) = ζ ai = ζ bi
and moreover the paths W s , s ∈ [a i ,b i ] coincide up to their exit time from D.
For every i ∈ I, we define a random element W (i) of Ω by setting for every s ≥ 0
W (i)
s (t) = W (ai +s)∧b i
(ζ ai + t),
for 0 ≤ t ≤ ζ (W
(i)
s ) := ζ (a i +s)∧b i
− ζ ai .
Informally, the W (i) ’s represent the excursions of the Brownian snake outside D (the word
“outside” is a bit misleading since these excursions may come back into D even though they
start from the boundary of D).
Finally, we also need a process that contains the information given by the Brownian snake
paths before they exit D. We set ˜W s D = W η D s
, where for every s ≥ 0,
∫ r
}
ηs {r D := inf ≥ 0 : du½{τ(W u)≥ζ u} > s .
0
The σ-field E D is by definition generated by the process ˜W D and by the class of N x -negligible
subsets of Ω (the point x is fixed throughout this subsection). The following statement is proved
in [36] (Proposition 2.3 and Theorem 2.4).
Theorem 3.2.4. There exists a random finite measure denoted by Z D , which is E D -measurable
and N x a.e. supported on ∂D, such that the following holds. Under N x , conditionally on E D , the
point measure
N := ∑ i∈I
δ (W (i) )
is Poisson with intensity ∫ ∂D ZD (dy) N y (·).
62

We will apply this theorem to the case d = 1, x = 0 and D =]c, ∞[ for some c < 0. In that
case, the measure Z D is a random multiple of the Dirac measure at c : Z D = L c δ c for some
nonnegative random variable L c . From Lemma 3.2.1(i) and Theorem 3.2.4, it is easy to verify
that {L c > 0} = {R∩] − ∞,c] ≠ ∅} = {R∩] − ∞,c[≠ ∅} , N 0 a.e. Moreover, as a simple
consequence of the special Markov property, the process (L −r ) r>0 is a nonnegative martingale
under N 0 (it is indeed a critical continuous-state branching process). In particular the variable
is finite N 0 a.e., for every r < 0.
L ∗,r :=
sup L c
c∈Q∩]−∞,r]
3.2.5. Uniqueness of the minimum. From now on we assume that d = 1. In this subsection,
we consider the Brownian snake under its excursion measure N 0 . We use the notation
W = inf Ŵ s .
s≥0
Note that the law of W under N 0 is given by Lemma 3.2.1(i) and an obvious translation argument.
Proposition 3.2.5. There exists N 0 a.e. a unique instant s ∗ ∈]0,σ[ such that Ŵs ∗
= W.
This result already appears as Lemma 16 in [45], where its proof is attributed to T. Duquesne.
We provide a short proof for the sake of completeness and also because this result plays a major
role throughout this work.
Proof : Set
{
}
λ := inf s ≥ 0 : Ŵs = W ,
{
}
ρ := sup s ≥ 0 : Ŵs = W
so that 0 < λ ≤ ρ < σ. We have to prove that λ = ρ. To this end we fix δ > 0 and we verify
that N 0 (ρ − λ > δ) = 0.
Fix two rational numbers q < 0 and ε > 0. We first get an upper bound on the quantity
N 0 (q − ε ≤ W < q , ρ − λ > δ).
Denote by (W (i) ) i∈I the excursions of the Brownian snake outside ]q, ∞[, and by N the corresponding
point measure, as in the previous subsection. Since the law of W under N 0 has no
atoms, the numbers W (i) , i ∈ I are distinct N 0 a.e. Therefore, on the event {W < q}, the whole
interval [ρ,λ] must be contained in a single excursion interval below level q. Hence,
}
N 0 (q − ε ≤ W < q , ρ − λ > δ) ≤ N 0
({∀i ∈ I : W (i) ≥ q − ε ∩
{
∃i ∈ I : σ
Introduce the events A ε := {W < q − ε} and B ε,δ = {W ≥ q − ε , σ > δ}. We get
N 0 (q − ε ≤ W < q , ρ − λ > δ) ≤ N 0 (N(A ε ) = 0, N(B ε,δ ) ≥ 1).
(
W (i)) > δ
From the special Markov property (and the remarks of the end of subsection 3.2.3), we know
that conditionally on L q , N is a Poisson point measure with intensity L q N q . Since the sets A ε
and B ε,δ are disjoint, independence properties of Poisson point measures give
)
N 0 (q − ε ≤ W < q , ρ − λ > δ) ≤ N 0
(½{N(A ε)=0} (1 − exp(−L q N q (B ε,δ )))
)
= N 0
(½{q−ε≤W

where c(ε,δ) = N q (B ε,δ ) does not depend on q by an obvious translation argument.
We can apply the preceding bound with q replaced by q − ε,q − 2ε, etc. By summing the
resulting estimates we get
)
N 0 (W < q , ρ − λ > δ) ≤ N 0
(½{W δ) = 0.
3.2.6. Bessel processes. Throughout this work, (ξ t ) t≥0 will stand for a linear Brownian
motion started at x under the probability measure P x . The notation ξ[0,t] will stand for the
range of ξ over the time interval [0,t]. For every δ > 0, (R t ) t≥0 will denote a Bessel process
of dimension δ started at x under the probability measure P x (δ) . We will use repeatedly the
following simple facts. First, if λ > 0, the process R (λ)
t := λ −1 R λ 2 t is under P x
(δ) a Bessel process
of dimension δ started at x/λ. Secondly, if 0 ≤ x ≤ x ′ and t ≥ 0, the law of R t under P x
(δ)
is stochastically bounded by the law of R t under P (δ)
x
. The latter fact follows from standard
′
comparison theorems applied to squared Bessel processes.
Absolute continuity relations between Bessel processes, which are consequences of the Girsanov
theorem, were first observed by Yor [55]. We state a special case of these relations, which
will play an important role in this work. This special case appears in Exercise XI.1.22 of [50].
Proposition 3.2.6. Let t > 0 and let F be a nonnegative measurable function on C([0,t], R).
Then, for every x > 0 and λ > 0,
∫ t
) ]
E x
[½{ξ[0,t]⊂]0,∞[} exp
(− λ2 dr
[
]
2 0 ξr
2 F((ξ r ) 0≤r≤t ) = x ν+1 2 E
(2+2ν)
x (R t ) −ν−1 2 F((Rr ) 0≤r≤t ) ,
√
where ν = λ 2 + 1 4 .
We shall be concerned by the case when λ 2 /2 = 6, and then 2 + 2ν = 9 and ν + 1/2 = 4.
Taking F = 1 in that case, we see that
(3.2.4) x 4 E x
(9) [ ]
R
−4
t ≤ 1.
3.3. Conditioning and re-rooting of trees
This section contains the proof of Theorem 3.1.1 and Theorem 3.1.2 which were stated in the
introduction. Both will be derived as consequences of Theorem 3.3.1 below. Recall the notation
s ∗ for the unique time of the minimum of Ŵ under N 0, and W [s] for the snake re-rooted at s.
Theorem 3.3.1. Let ϕ : R + −→ R + be a continuous function such that ϕ(s) ≤ C(1 ∧ s) for
some finite constant C. Let F : Ω −→ R + be a bounded continuous function. Then,
(
)
lim
ε→0 ε−4 N 0 σ ϕ(σ)F(W)½{W >−ε} = 2 ( (
21 N 0 ϕ(σ)F W [s∗])) .
□
64

The proof of Theorem 3.3.1 occupies most of the remainder of this section. This proof will
depend on a series of lemmas. To motivate these lemmas, we first observe that, from the rerooting
identity in Theorem 3.2.3, we have
)
∫ σ (
)
N 0
(σ ϕ(σ)F(W)½{W >−ε} = N 0
(ϕ(σ) ds F W [s])½{W[s] >−ε}
0
∫ σ (
)
(3.3.1)
= N 0
(ϕ(σ) ds F W [s])½{ W c s

Similarly,
N 0
(½{W>−ε}e −σ ∫ σ
=
=
=
∫ δ
0
∫ δ
0
∫ δ
0
0
ds½{ζ s−ε} exp
)
(
−4
(
daE ε
[½{ξ a
>0} exp −4
∫ a
0
∫ a
∫ a
daE ε
[½{ξ a
>0} exp
(−2 3/2 dt
0
)]
dt N ξt (1 −½{W>−ε)}e −σ )
)]
dt N ξt (1 −½{W>0)}e −σ )
0
( ) ))] 2
3coth
(2 1/4 ξ t − 2
using Lemma 3.2.1(ii) in the last equality. For every x > 0, set
h(x) = − 3 ( ( ) ) 2
2x 2 + 2−1/2 3coth 2 1/4 x − 2 > 0.
A Taylor expansion shows that h(x) ≤ C x 2 . (Here and later, C,C ′ ,C ′′ denote constants whose
exact value is unimportant.) Then,
∫ σ
) ∫ δ
( ∫ a ∫
N 0
(½{W>−ε}e −σ dt a
)]
= daE ε
[½{ξ a
>0} exp −6 − 4 dt h(ξ t )
0
ds½{ζ s−ε}(1 − e −σ )
=
≤
∫ δ
0
4C
∫ σ
0
ds½{ζ s−ε}(1 − e −σ ) ds½{ζ s

Lemma 3.3.3. For every δ > 0,
( ( ∫ σ
) ) 2
sup N 0 ε −4 ds½{ W
0

To simplify notation, we have written E instead of E a,b,c , and η[0,a] obviously denotes the set
{η t : 0 ≤ t ≤ a}, with a similar notation for η ′ [0,b] and η ′′ [0,c].
In the preceding formula for Jε
a,b,c
quantity depending on y = (η
b ′ ∧ η′′
(3.3.9)
J a,b,c
ε
E
[½{η[0,a]⊂]−y,∞[} exp
(
−6
c
∫ a
0
, conditioning with respect to the pair (η ′ ,η ′′ ) leads to a
) + ε, of the form
)]
( ∫
dt
a
)]
dt
(η t + y) 2 = E y
[½{ξ[0,a]⊂]0,∞[} exp −6
0
= y 4 E (9)
y
[ ]
R
−4
a
using Proposition 3.2.6 as in the proof of Lemma 3.3.2 above. Hence,
= E
[½{|η b ′ − η′′ c | < ε,η′ [0,b] ⊂] − ε, ∞[,η ′′ [0,c] ⊂] − ε, ∞[}
( (∫
× ((η b ′ ∧ η′′ c ) + ε)4 E (9) [ ] b
(η R
−4
b ′ ∧η′′ c )+ε a exp −6
0
∫
dt c
(η t ′ + + ε)2
0
ξ 2 t
dt
(η ′′
t + ε)2 )) ] .
Recall that our goal is to bound ∫ dadbdc½{a+b≥δ,a+c≥δ} Jε
a,b,c . First consider the integral over
the set {a < δ/2}. Then plainly we have b > δ/2 and c > δ/2, and we can use (3.2.4) to bound
∫
{a 0,
∫ ∞
∫
(3.3.10)
dbE y
(9) [ ] ∞
R
−4
b ≤ dbE (9) [ ]
0 R
−4
b = C
′
δ < ∞.
δ/2
0
))]
dt
(η t ′′ + ε)2
We still have to get a similar bound for the integral over the set {a ≥ δ/2}. Applying the
bound (3.3.10) with y = (η
b ′ ∧ η′′ c ) + ε, we see that it is enough to prove that
∫ [½{|η b ′ − η′′ c | < ε, η′ [0,b] ⊂] − ε, ∞[, η ′′ [0,c] ⊂] − ε, ∞[}
(3.3.11)
[0,∞[ 2 dbdcE
( (∫ b
×((η b ′ ∧ η′′ c ) + ε)4 exp −6
0
∫
dt c
(η t ′ + + ε)2
From Proposition 3.2.6 again, the left-hand side of (3.3.11) is equal to
∫
(
(3.3.12) ε 8 dbdcE ε (9) ⊗ E ε
[½{|R (9)
[0,∞[ 2 b −R e c|

where R and ˜R are two independent nine-dimensional Bessel processes started at ε under the
probability measure P ε (9) ⊗ P ε
(9) . The quantity (3.3.12) is bounded above by ε 8 (I1 ε + Iε 2 ), where
∫
(
I1 ε = dbdcE ε (9) ⊗ E ε
[½{R (9)
b

N = ∑ i∈I δ ω i with intensity xN 0, under the probability measure P (x) . To simplify notation, we
write Ws i = W s (ω i ). We then set
( )
W = inf inf Ŵs
i = inf W i .
i∈I s≥0 i∈I
Lemma 3.3.5. For every x > 0 and ε > 0,
E (x)
[ ∑
i∈I
∫ σ(ω i )
ds½{ W
0
c s

which gives the formula of the lemma, with
[∫ ∞
(
g(u) = E u
(9) dbR −4
b
exp − 3 )]
0
2Rb
2 .
The fact that g is nondecreasing follows from the strong Markov property of R. The continuity
of g is easy from a similar argument. Finally, the value of g(0) is obtained from the explicit
formula for the Green function of nine-dimensional Brownian motion.
□
Recall our notation E ]a,∞[ for the σ-field generated by the Brownian snake paths before their
first exit from ]a, ∞[, and L a for the total mass of the exit measure Z ]a,∞[ .
Lemma 3.3.6. Let a < 0. For every bounded E ]a,∞[ -measurable function Φ on Ω,
)
lim N 0
= 2
ε→0 21 N 0
(½{W

It remains to study
ε −4 N 0
(½{W

and with a suitable choice of Φ), we get, for every integer n ≥ n 0 and every i ∈ Z such that
−2 n ≤ i2 −n < a,
( (
lim ε −4 i2
N 0
(½{W

Proof of Theorems 3.1.1 and 3.1.2 : Both Theorems 3.1.1 and 3.1.2 follow from the
convergence
( )
(3.3.20) lim ε −4 N (1)
ε→0
0 F(W)½{W>−ε} = 2 ( (
21 N(1) 0 F W [s∗])) ,
which holds for every bounded continuous function F on Ω = C(R + , W) (take F = 1 to recover
the first assertion of Theorem 3.1.1). We will now derive (3.3.20) from Theorem 3.3.1.
For every λ > 0, let us introduce the scaling operator θ λ defined on Ω by
ζ s ◦ θ λ = λ 1/2 ζ s/λ ,
W s ◦ θ λ (t) = λ 1/4 W s/λ (λ −1/2 t) .
Note that, for every r > 0, the image of N (r)
0 under θ 1/r is N (1)
0 .
Let δ ∈]0,1[. It follows from Theorem 3.3.1 that the law of the pair (σ,W) under
µ ε,δ := ε −4 N 0 (· ∩ {W > −ε, 1 − δ < σ < 1})
converges weakly as ε → 0 towards the law of (σ,W [s∗] ) under the measure µ δ having density
2/(21σ) with respect to N 0 (· ∩ {1 − δ < σ < 1}).
Since the mapping (r,ω) → θ r ω is continuous, it follows that the law of W ◦ θ 1/σ under µ ε,δ
converges as ε → 0 towards the law of W [s∗] ◦ θ 1/σ under µ δ . Thus,
lim µ ε,δ(F(W ◦ θ 1/σ )) = µ δ (F(W [s∗] ◦ θ 1/σ )),
ε→0
or equivalently
) ( 2
lim
ε→0 ε−4 N 0
(½{W>−ε, 1−δ

Proposition 3.3.7. For any bounded continuous function F on Ω,
lim N (1)
0 ε↓0
(F(W) | Z(] − ∞,0]) ≤ ε) = N(1) 0 (F).
Proof : From the re-rooting theorem 3.2.3 and the remark after this statement we have,
(∫
(3.3.21) N (1) ( 1 (
)
(1)
0 F(W)½{Z(]−∞,0])≤ε})
= N 0 ds F W [s])½{Z(]−∞, W c s])≤ε}
.
Now, since the measure Z has no atoms,
∫ 1
∫
ds½{Z(]−∞, W c = s])≤ε}
0
0
Z(dy)½{Z(]−∞,y])≤ε} = ε.
In particular by taking F = 1 in (3.3.21) we see that the law of Z(]−∞,0]) under N (1)
0 is uniform
over [0,1] (this fact was already observed in Section 3.2 of [6]). Hence,
N (1)
0 (F(W) | Z(] − ∞,0]) ≤ ε) = ε−1 N (1)
0
(∫ 1
ds F
0
(
W [s])½{Z(]−∞, c W s])≤ε}
On the other hand the closed sets {s ∈ [0,1] : Z(] − ∞,Ŵs]) ≤ ε} decrease to the singleton
{s ∗ } as ε ↓ 0, N (1)
0
a.e. It follows that
∫ 1
lim ε −1 ds F
(W [s])½{Z(]−∞, c ε↓0 W = F s])≤ε}
(W [s∗])
0
N (1)
0 a.e. Using dominated convergence and Theorem 3.1.2, we get Proposition 3.3.7. □
Remark. Up to some point, the approximation given in Proposition 3.3.7 may seem as
“reasonable” as the one we used in Theorem 3.1.1 to define N (1)
0 . In applications such that the
ones developed in [39], the approximation given by Theorem 3.1.1 turns out to be more useful.
3.4. Other conditionings
Motivated by Theorem 3.3.1, we define a σ-finite measure N 0 on Ω by setting
0
N 0 (F) = N 0
( 1
σ F (W [s∗])) .
Theorem 3.3.1 shows that, up to the multiplicative constant 2/21, N 0 is the limit in an appropriate
sense of the measures ε −4 N 0 (· ∩ {W > −ε}) as ε → 0. We have also
∫ ∞ ( (
N 0 (F) =
F W [s∗])) ∫ ∞
=
dr
2 √ 2πr 5 N(r) 0
0
dr
2 √ 2πr 5 N(r) 0 (F),
where N (r)
0 can be defined equivalently as the law of W [s∗] under N (r)
0 , or as the image of N(1) 0
under the scaling operator θ r .
We will now describe a different approach to N 0 , which involves conditioning the Brownian
snake excursion on its height H = sup s≥0 ζ s , rather than on its length as in Theorem 3.1.1. This
will give more insight in the behavior of the Brownian snake under N 0 . Eventually, this will lead
to a construction of a Brownian snake excursion with infinite length conditioned to stay on the
positive side. We rely on some ideas from [1].
)
.
75

For every h > 0, we set N h 0 = N 0(· | H = h). Then,
N 0 =
∫ ∞
0
dh
2h 2 Nh 0 .
From Theorem 1 in [1] we know that there exists a constant c 0 > 0 such that
(3.4.1) lim
ε→0
ε −4 N 1 0(W > −ε) = c 0 .
A simple scaling argument then implies that, for every h > 0,
(3.4.2) lim
ε→0
ε −4 N h 0(W > −ε) = c 0
h 2.
Theorem 3.4.1. For every h > 0, there exists a probability measure N h 0 on Ω such that
lim
ε→0 Nh 0(· | W > −ε) = N h 0
in the sense of weak convergence on the space of probability measures on Ω. Moreover,
N 0 = 21c 0
4
∫ ∞
0
dh
h 4 Nh 0.
Remark. Our proof of the first part of Theorem 3.4.1 does not use Section 3.3. This proof
thus gives another approach to the conditioned measure N 0 , which does not depend on the
re-rooting method that played a crucial role in Section 3.3.
Before proving Theorem 3.4.1, we will establish an important preliminary result. We first
introduce some notation. Following [1], we set for every ε > 0,
and, for every x > 0,
f(ε) = N 1 0 (W > −ε)
G(x) = 4
∫ x
0
u(1 − f(u))du.
The function G is obviously nondecreasing. It is also bounded since
G(∞) = 4
∫ ∞
0
u N 1 0(W ≤ −u)du = 2
by a scaling argument and Lemma 3.2.1(i).
∫ ∞
0
r −2 N r 0(W ≤ −1)dr = 4 N 0 (W ≤ −1) = 6
By well-known properties of Brownian excursions, there exists N h 0 a.s. a unique time α ∈]0,σ[
such that ζ α = h. The next proposition discusses the law of W α under N h 0 (· | W > −ε).
Proposition 3.4.2. Let Φ be a bounded continuous function on W. Then,
[
(∫
lim ε −4 N h ( h
( ))]
(9)
0 Φ(Wα )½{W>−ε})
= E 0 Φ(R t ,0 ≤ t ≤ h)R −4 dt Rt
ε↓0
h
exp G √ . h − t
Remarks. (i) From the bound G(x) ≤ 6 ∧ (2x 2 ), it is immediate to verify that
∫ h
( )
dt Rt
G √ < ∞ , P (9)
h − t
0 a.s.
0
R 2 t
(ii) By taking Φ = 1, we see that the constant c 0 in (3.4.1) is given by
[ (∫ 1
( ))]
c 0 = E (9)
0 R1 −4 dt Rt
exp G √ , 1 − t
0
R 2 t
0
R 2 t
76

as it was already observed in [1]. The fact that the quantity in the right-hand side is finite
follows from the proof below.
Proof : Our main tool is Williams’ decomposition of the Brownian excursion at its maximum
(see e.g. Theorem XII.4.5 in [50]). For every s ≥ 0, we set
ρ s = ζ s∧α ,
ρ ′ s = ζ (σ−s)∨α.
Under the probability measure N h 0 , the processes (ρ s) s≥0 and (ρ ′ s) s≥0 are two independent threedimensional
Bessel processes started at 0 and stopped at their first hitting time of h.
We also need to introduce the excursions of ρ and ρ ′ above their future infimum. Set
ρ s
= inf
r≥s ρ r
and let (a j ,b j ), j ∈ J be the connected components of the open set {s ≥ 0 : ρ s > ρ s
}. For every
j ∈ J, define
Then, by excursion theory,
ζ j s = ρ (aj +s)∧b j
− ρ aj , s ≥ 0
h j = ρ aj .
∑
δ (hj ,ζ j )(dr de)
is a Poisson point measure on R + × C(R + , R + ) with intensity
j∈J
2½[0,h](r)½[0,h−r](H(e))dr n(de)
where H(e) = sup s≥0 e(s) as previously. The same result obviously holds for the analogous point
measure
∑
j∈J ′ δ (h ′
j ,ζ ′j )(dr de)
obtained by replacing ρ with ρ ′ .
We can combine the preceding assertions with the spatial displacements of the Brownian
snake, in a way very similar to the proof of Lemma V.5 in [37]. For every j ∈ J, we set
W j s (t) = W (aj +s)∧b j
(h j + t) − Ŵa j
, 0 ≤ t ≤ ζ j s, s ≥ 0.
Note that by the properties of the Brownian snake Ŵa j
= Ŵb j
= W α (h j ). Then,
N := ∑ j∈J
δ (hj ,W j )
is under N h 0 a Poisson point measure on R + × Ω, with intensity
(3.4.3) 2½[0,h](r)½[0,h−r](H(ω))dr N 0 (dω).
The same holds for the analogous point measure
N ′ := ∑ j∈J ′ δ (h ′
j ,W ′j ).
Moreover N and N ′ are independent and the pair (N,N ′ ) is independent of W α . All these
assertions easily follow from properties of the Brownian snake.
77

Now note that the range of the Brownian snake under N h 0 can be decomposed as
( ⋃
) ( ⋃
)
{W α (t) : 0 ≤ t ≤ h} ∪ (W α (h j ) + R(W j )) ∪ (W α (h ′ j) + R(W ′j )) .
j∈J
j∈J ′
Using this observation and conditioning with respect to W α , we get
(
Φ(Wα )½{W>−ε})
N h 0
= N h 0
(
(
Φ(W α )½{W α(t)>−ε, 0≤t≤h} exp −4
= E 0
[Φ(ξ t , 0 ≤ t ≤ h)½{ξ[0,h]⊂]−ε,∞[} exp
Then, for every x > −ε,
N x (H < h − t, W ≤ −ε) =
and we obtain
(3.4.4)
0
N h 0 (Φ(W α)½{W>−ε})
∫ h−t
= E 0
[Φ(ξ t , 0 ≤ t ≤ h)½{ξ[0,h]⊂]−ε,∞[} exp
0
∫ h
0
(
−4
))
dt N Wα(t)(H < h − t, W ≤ −ε)
∫ h
0
)]
dt N ξt (H < h − t, W ≤ −ε) .
∫
du
h−t
( ( ))
2u 2 Nu x (W ≤ −ε) = du x + ε
2u 2 1 − f √ u
(
−2
∫ h
(
= E ε
[Φ(ξ t − ε, 0 ≤ t ≤ h)½{ξ[0,h]⊂]0,∞[} exp −2
0
∫ h
0
∫ h−t
dt
0
0
∫ h−t
dt
0
(
du
u 2 1 − f
(
ξt + ε
√ u
)))]
( ( )))]
du ξt
u 2 1 − f √ . u
For every x > 0, the change of variable v = x/ √ u gives
∫ h−t
( ( )) ∫
du x
∞
(
u 2 1 − f √u = 2x −2 dv v(1 − f(v)) = x −2 3 − 1 ( )) x
2 G √ . h − t
x/ √ h−t
By substituting this into (3.4.4) and using Proposition 3.2.6 once more, we get
(
Φ(Wα )½{W>−ε})
N h 0
= E ε
[Φ(ξ t − ε, 0 ≤ t ≤ h)½{ξ[0,h]⊂]0,∞[} exp
( ∫ h
−6
[
(∫ h
( ))]
= ε 4 E ε
(9) Φ(R t − ε,0 ≤ t ≤ h)R −4 dt Rt
(3.4.5)
h
exp
0 Rt
2 G √ . h − t
In view of (3.4.5), the proof of Proposition 3.4.2 reduces to checking that
(3.4.6)
lim E ε
(9)
ε↓0
= E (9)
0
[
Φ(R t − ε,0 ≤ t ≤ h)R −4
h
exp
[
Φ(R t ,0 ≤ t ≤ h)R −4
h
exp
(∫ h
0
(∫ h
0
0
dt
R 2 t
dt
ξ 2 t
dt
R 2 t
G
∫ h
( ))]
dt ξt
+
0 ξt
2 G √
h − t
G
(
Rt
√
h − t
))]
(
Rt
√
h − t
))]
This follows from a dominated convergence argument, which at the same time will prove that the
quantity in the right-hand side of (3.4.6) is well-defined. Note that we may define on a common
probability space, a nine-dimensional Bessel process X ε = (Xt ε ,t ≥ 0) started at ε, for every
ε ≥ 0, in such a way that the inequality X ε ≥ X 0 holds a.s. for every ε > 0. Since
( ) X
ε
G √ t
≤ 4(Xε t ) 2
, ∀t ∈ [0,h/2]
h − t h
78

we first get
(∫ h
( ))
(Xh ε dt X
ε )−4 exp
(Xt ε G √ t
)2 h − t
(3.4.7)
0
≤
≤
e 2 (X ε h )−4 exp
( ∫ h
e 2 ( (
Xh
0 ) −4
exp 6
h/2
∫ h
( ) )
dt X
ε
(Xt ε G √ t
)2 h − t
h/2
)
dt
(Xt 0 ,
)2
using the bounds G ≤ 6 and X ε ≥ X 0 . Then, an application of Itô’s formula shows that
( ∫ )
t
(Xt 0 ) −4 dr
exp 6
(Xr 0)2 is a local martingale on the time interval [h/2, ∞[, and so
[ ( ∫ )]
(X ) h
0 −4 dt
[
E h exp 6
(Xt 0 ≤ E (X 0
)2 h/2 )−4] < ∞.
h/2
Together with (3.4.7), this shows that the random variables appearing in the left-hand side of
(3.4.6) are uniformly integrable. The convergence (3.4.6) easily follows. □
Proof of Theorem 3.4.1 : We first explain how the first part of Theorem 3.4.1 can be
deduced from Proposition 3.4.2. Recall the notation (N,N ′ ) from the proof of this proposition.
We first observe that we can find a measurable functional Γ such that
h/2
W = Γ(W α , N,N ′ ) ,
Let us make this functional more explicit. We have first
α = ∑ j∈J
σ(W j ).
N h 0 a.s.
For every l ∈ [0,h], we set
τ l = ∑ j∈J½{h j ≤l} σ(W j ).
Then, if s ∈ [0,α], there is a unique l such that τ l− ≤ s ≤ τ l , and :
• Either there is a (unique) j ∈ J such that l = h j , and
• Or there is no such j, and
ζ s = l + ζ j s−τ l−
,
{
Wα (t) if t ≤ l ,
W s (t) =
W α (l) + W j s−τ l−
(t − l) if l < t ≤ ζ s ;
ζ s = l ,
W s (t) = W α (t) , t ≤ l .
The previous formulas identify (W s ,0 ≤ s ≤ α) as a measurable function of the pair (W α , N),
and in a similar way we can recover (W σ−s ,0 ≤ s ≤ σ − α) as the same measurable function of
(W α , N ′ ).
To simplify notation, write N h,(ε) for the conditional probability N h 0 (· | W > −ε). From
elementary properties of Poisson measures, we get that under the probability measure N h,(ε)
0
79

and conditionally given W α , the point measures N and N ′ are independent and Poisson with
intensity
µ h ε(W α ;dr dω) := 2½[0,h](r)½[0,h−r](H(ω))½{R(ω)⊂]−ε−W α(r),∞[} dr N 0 (dω).
As a consequence of Proposition 3.4.2, the law of W α under N h,(ε)
0 converges as ε → 0 to the law
of the process Y h = (Yt h ,0 ≤ t ≤ h) such that
[
(3.4.8) E Φ
(Y h)] [
(∫ h
( ))]
= h2
E (9)
0 Φ(R t ,0 ≤ t ≤ h)R −4 dt Rt
c
h
exp G √ .
0 h − t
Suppose that on the same probability space where Y h is defined, we are also given two random
point measures M and M ′ on R + × Ω, which conditionally given Y h are independent Poisson
point measures with intensity
(3.4.9) µ h 0 (Y h ;dr dω) := 2½[0,h](r)½[0,h−r](H(ω))½{R(ω)⊂]−Yr h ,∞[} dr N 0(dω).
From the continuity properties of the “reconstruction mapping” Γ, it should now be clear that
the probability measures N h,(ε) converge as ε → 0 to the measure N h 0 defined as the law of
Γ(Y h , M, M ′ ). Here we leave some easy technical details to the reader.
Let us prove the second assertion of Theorem 3.4.1. Let us fix s 1 > 0, and let ψ be a
continuous function on R + with compact support contained in ]0, ∞[. Let F be a bounded
continuous function on Ω. It follows from Theorem 3.3.1 that
(
lim
ε→0 ε−4 N 0 ψ(ζ s1 )F(W)½{W>−ε}
)
= 2
21 N 0
= 2
(
σ −1 ψ
0
R 2 t
( )
ζ s [s∗]
1
F
(W [s∗]))
(3.4.10)
21 N 0(ψ(ζ s1 )F(W)).
To see this, apply Theorem 3.3.1 with a function ϕ such that sϕ(s) vanishes on a neighborhood
of 0 and is identically equal to 1 on [s 1 , ∞[.
On the other hand, we have also
) ∫ ∞
ε −4 N 0
(ψ(ζ s1 )F(W)½{W>−ε} = ε −4 dh
(
)
2h 2 Nh 0 ψ(ζ s1 )F(W)½{W >−ε}
(3.4.11)
=
∫ ∞
0
0
dh
2h 2 ε−4 N h 0 (W > −ε) × Nh,(ε) 0 (ψ(ζ s1 )F(W)).
We pass to the limit ε → 0 in the right-hand side of (3.4.11), using (3.4.2) and the first assertion
of Theorem 3.4.1, which gives
lim
ε→0 Nh,(ε) 0 (ψ(ζ s1 )F(W)) = N h 0 (ψ(ζ s 1
)F(W)).
To justify dominated convergence, first note that
(3.4.12) ε −4 N h 0 (W > −ε) = ε−4 N 1 0
(
W > − ε √
h
)
≤ C h 2.
Furthermore, by comparing the intensity measures in (3.4.3) and (3.4.9), we get that the distribution
of σ under N h,(ε)
0 is stochastically bounded by the distribution of σ under N h 0
(
. )
Hence,
N h,(ε)
0 (ψ(ζ s1 )F(W)) ≤ C ′ N h,(ε)
0 (σ > s 1 ) ≤ C ′ N h 0(σ > s 1 ) ≤ C (s1 ) exp − C′ (s 1 )
h 2 ,
where C (s1 ) and C ′ (s 1 ) are positive constants depending on s 1.
80

The previous observations allow us to apply the dominated convergence theorem to the righthand
side of (3.4.11), and to get
(
)
lim ε −4 N 0 ψ(ζ s1 )F(W)½{W>−ε} = c ∫ ∞
0 dh
ε↓0 2 h 4 Nh 0 (ψ(ζ s 1
)F(W)).
Comparing with (3.4.10) now completes the proof.
At this point we have obtained two distinct descriptions of N 0 :
• The law of σ under N 0 has density (8π) −1/2 s −5/2 , and the conditional distribution
N 0 (· | σ = s) is the law under N (s)
0 of the re-rooted snake W [s∗] .
• The law of H under N 0 has density 21c 0
4
h −4 , and the conditional distribution N 0 (· |
H = h) can be reconstructed from the “spine” Y h and the Poisson point measures M
and M ′ as explained in the proof of Theorem 3.4.1.
If we think of analogous results for the Itô measure of Brownian excursions, it is tempting
to look for a more Markovian description of N 0 . It is relatively easy to see that the process
((ζ s ,W s ),s > 0) is Markovian under N 0 , and to describe its transition kernels (informally, this
is the Brownian snake conditioned not to exit ]0, ∞[ – compare with [2]). One would then like
to have an explicit formula for entrance laws, that is for the law of (ζ s ,W s ) under N 0 , for each
fixed s > 0. Such explicit expressions seem difficult to obtain. See however the calculations in
Section 3.5.
In the final part of this section, we investigate the limiting behavior of the measures N 0 (· |
H = h) as h → ∞. This leads to a (one-dimensional) Brownian snake conditioned to stay
positive and to live forever. The motivation for introducing such a process comes from the fact
that it is expected to appear in scaling limits of discrete trees coding random quadrangulations :
See the recent work of Chassaing and Durhuus [13].
Before stating our result, we give a description of the limiting process. Let Z = (Z t ,t ≥ 0)
be a nine-dimensional Bessel process started at 0. Conditionally given Z, let
P = ∑ i∈I
be a Poisson point measure on R + × Ω with intensity
δ (hi ,ω i )
2½{R(ω)⊂]−Z r,∞[} dr N 0 (dω).
We may and will assume that P is constructed in the following way. Start from a Poisson point
measure
Q = ∑ δ (hj ,ω j )
j∈J
with intensity 2 dr N 0 (dω), and assume that Q is independent of Z. Then set
P = ∑ j∈J½{R(ω j )⊂]−Z hj ,∞[} δ (hj ,ω j ) .
0
□
We then construct our conditioned snake W ∞ from the pair (Z, P). This is very similar to
the reconstruction mapping that was already used in the proof of Theorem 3.4.1. To simplify
notation, we put
σ i = σ(ω i ) , ζ i s = ζ s(ω i ) , W i s = W s(ω i )
81

for every i ∈ I and s ≥ 0. For every l ∈ [0,h], we set
τ l = ∑ i∈I½{h i ≤l} σ i .
Then, if s ≥ 0, there is a unique l such that τ l− ≤ s ≤ τ l , and :
• Either there is a (unique) i ∈ I such that l = h i , and we set
ζs ∞ = l + ζs−τ i l−
,
{
Ws ∞ Zt if t ≤ l ,
(t) =
Z l + Ws−τ i l−
(t − l) if l < t ≤ ζs ∞ ;
• Or there is no such i, and we set
ζ ∞ s = l ,
W ∞ s (t) = Z t , t ≤ l .
It is easy to verify that these prescriptions define a continuous process W ∞ with values in W.
We denote by N ∞ 0 the law of W ∞ .
Theorem 3.4.3. The probability measures N h 0 converge to N ∞ 0 when h → ∞.
Proof : We rely on the explicit description of N h 0 obtained in the proof of Theorem 3.4.1.
Let Y h = (Y h
t ,0 ≤ t ≤ h) be as in (3.4.8).
Lemma 3.4.4. The processes (Yt∧h h ,t ≥ 0) converge in distribution to Z as h → ∞.
Proof : Let A > 0 and let Φ be a bounded continuous function on C([0,A], R + ). By (3.4.8),
if h ≥ A,
[
] [
(∫ h
( ))]
E Φ(Yt h ,0 ≤ t ≤ A) = h2
E (9)
0 Φ(R t ,0 ≤ t ≤ A)R −4 dt Rt
c
h
exp
0 0 Rt
2 G √ . h − t
We apply the Markov property at time A in the right-hand side, and write h = A+a to simplify
notation :
[
[
] (∫ A
E Φ(Yt h ,0 ≤ t ≤ A) = h2
E (9)
dt
0 Φ(R t ,0 ≤ t ≤ A) exp
c 0
(3.4.13)
From the bound 0 ≤ G(x) ≤ 2x 2 , it is immediate that
(∫ A
dt
(3.4.14) 1 ≤ exp G
0
R 2 t
On the other hand, a scaling argument gives
h 2 [ (∫ a
E (9)
c
R A
Ra
−4 dt
exp
0 0 Rt
2 G
( ) h 2
= c −1
0 E (9)
a R A / √ a
0
R 2 t
[ (∫ a
× E (9)
R A
Ra
−4 exp
(
Rt
√
h − t
))
≤
(
Rt
√ a − t
))]
[ (∫ 1
R1 −4 exp
0
dt
R 2 t
0
(
1 + A a
) 2
.
( ))
Rt
G √
h − t
dt
R 2 t
( ))]
Rt
G √ . 1 − t
( ))] ]
Rt
G √ . a − t
82

From (3.4.6), we know that
(3.4.15)
[ (∫ 1
lim E x
(9) R1 −4 dt
exp
x↓0 0
R 2 t
( ))] [ (∫ 1
Rt
G √ = E (9)
1 − t
0 R1 −4 exp
0
dt
R 2 t
( ))]
Rt
G √ = c 0 .
1 − t
We can use (3.4.14) and (3.4.15) to pass to the limit h → ∞ in the right-hand side of (3.4.13).
The justification of dominated convergence is easy thanks to the bounds we obtained when
proving (3.4.6). It follows that
lim E[Φ(Y t h ,0 ≤ t ≤ A)] = E (9)
0 [Φ(R t,0 ≤ t ≤ A)]
h→∞
which was the desired result.
We can now complete the proof of Theorem 3.4.3. By Lemma 3.4.4 and the Skorokhod
representation theorem, we may assume that (Yt∧h h ) t≥0 converges to (Z t ) t≥0 uniformly on every
compact subset of R + , a.s.
Recall the description of N h 0 as the law of Γ(Y h , M, M ′ ) in the proof of Theorem 3.4.1 :
According to this description, we can construct a process (W h s ) s≤αh having the distribution of
(W s ) s≤α under N h 0, by the same formulas we used to define W ∞ from the pair (Z, P), provided
that Z is replaced by Y h , the point measure P is replaced by
N h := ∑ j∈J½{R(ω j )⊂]−Y h h j
,∞[}½{h j

and then, for every u ∈ T \{∅}, constructing ξ u as the solution of
⎧
⎪⎨ dξ u t = dξt u + 4
ξ u dt , 0 ≤ t ≤ h u ,
t
⎪⎩ ξ u 0 = ξ π(u)
h π(u)
.
We then define (V a ,a ∈ ˜θ) by the formula V pθ (u,l) = ξ u l for every u ∈ T and l ∈ [0,h u ]. Finally,
for every leaf a of ˜θ, we define the stopped path w (a) from (V a ,a ∈ ˜θ) in the same way as w (a)
was defined from (V a ,a ∈ ˜θ). Recall the notation L(θ) for the set of leaves of ˜θ, and I(θ) for the
set of its nodes.
Theorem 3.5.1. Let p ≥ 1 be an integer. Let F be a symmetric nonnegative measurable
function on W p . Then,
N 0
( ∫
]0,σ[ p ds 1 ...ds p F ( W s1 ,...,W sp
) )
∫
= 2 p−1 p!
⎡
( ( ) ∏
Λ p (dθ)Q θ 0
⎣F w (a)) a∈L(θ)
a∈I(θ)
(
V a
) 4
∏
a∈L(θ)
(
V a
) −4
⎤
Proof : We may assume that F is continuous and bounded above by 1, and that there exist
positive constants δ and M such that F(w 1 ,... ,w p ) = 0 as soon as ζ (wi ) /∈ [δ,M] for some i.
The proof will be divided in several steps.
Step 1. To simplify notation, we write R(V ) = {V a ,a ∈ ˜θ}, for the range of V , or equivalently
for the union of the ranges of w (a) for a ∈ L(θ). We first apply Theorem 3.2.2 to compute
)
N 0
( ∫
]0,σ[ p ds 1 ...ds p F ( W s1 ,...,W sp
)½{R⊂]−ε,∞[}
= p!2 p−1 ∫
= p!2 p−1 ∫
= p!2 p−1 ∫
[
(
Λ p (dθ)Q θ 0 F((w (a) ) a∈L(θ) )½{R(V )⊂]−ε,∞[} exp
[
Λ p (dθ)Q θ 0 F((w (a) ) a∈L(θ) )½{R(V )⊂]−ε,∞[} exp
( ( (
Λ p (dθ)Q θ ε F −ε + w (a)) a∈L(θ)
We then use Proposition 3.2.6 inductively to see that
[ ( (
ε −4 Q θ ε F −ε + w (a)) a∈L(θ)
⎡
( ( ) ∏
= Q θ ε
⎣F −ε + w (a)) a∈L(θ)
∫
− 4
∫
(
−6
)½{R(V )⊂]0,∞[} exp
(
−6
)½{R(V )⊂]0,∞[} exp
(
−6
a∈I(θ)
(
V a
) 4
∏
⎦.
)]
L θ (da) N 0 (r ⊂] − ε − V a , ∞[)
)]
L θ (da)
(V a + ε) 2
a∈L(θ)
∫ )]
Lθ (da)
(V a ) 2 .
∫ )]
Lθ (da)
(V a ) 2
(
V a
) −4
⎤
⎦ .
84

We have thus proved that
)
(3.5.1)
ε −4 N 0
( ∫
]0,σ[ p ds 1 ...ds p F ( W s1 ,... ,W sp
)½{W>−ε}
⎡
∫ ( (
= p!2 p−1 Λ p (dθ)Q θ ε
⎣F −ε + w (a)) ) ∏
a∈L(θ)
a∈I(θ)
⎤
( ) 4
∏ ( ) −4
V a V a
⎦.
a∈L(θ)
Step 2. We focus on the right-hand side of (3.5.1). Our goal is to prove that
⎡
⎤
∫ ( ( ) ∏
lim Λ p (dθ)Q θ ε
⎣F −ε + w (a)) ( ) 4
∏ ( ) −4
V a V a
⎦
ε→0
a∈L(θ)
a∈I(θ) a∈L(θ)
⎡
⎤
∫ ( ( ) ∏
= Λ p (dθ)Q θ 0
⎣F −ε + w (a)) ( ) 4
∏ ( ) −4
(3.5.2)
V a V a
⎦ .
a∈L(θ)
We first state a lemma.
Lemma 3.5.2. We have
⎡
Q θ ε
⎣ ∏
a∈I(θ)
where D(θ) = max{d θ (0,a) : a ∈ L(θ)}.
(
V a
) 4
∏
a∈L(θ)
(
V a
) −4
⎤
a∈I(θ)
⎦ ≤ E (9)
ε
[ ]
R −4
D(θ)
a∈L(θ)
Proof : We argue by induction on p. If p = 1, the result is immediate, with an equality. Let
p ≥ 2 and let us assume that the result holds at order 1,2,... ,p − 1. Let θ = (T ,(h u ,u ∈ T ))
be a marked tree with p leaves. Write h = h ∅ . By decomposing θ at its first branching point,
we get two marked trees θ ′ ∈ T j , and θ ′′ ∈ T p−j , for some j ∈ {1,... ,p − 1}, in such a way that
⎡
⎤
⎣ ∏ ( ) 4
∏ ( ) −4
V a V a
⎦
Q θ ε
a∈I(θ)
= E (9)
ε
⎡
a∈L(θ)
⎡
⎣R h 4 ⎣ ∏
Qθ′ R h
a∈I(θ ′ )
[ [ ]
≤ E ε
(9) Rh 4 E(9) R h
R −4
D(θ ′ )
(
V a
) 4
∏
a∈L(θ ′ )
[ ]]
E (9)
R h
R −4
D(θ ′′ )
(
V a
) −4
⎤
.
⎡
⎦Q θ′′ ⎣ ∏
R h
a∈I(θ ′′ )
(
V a
) 4
∏
a∈L(θ ′′ )
⎤⎤
( ) −4
V a
⎦⎦
We have used the induction hypothesis in the last inequality. We now observe that D(θ) =
h+max{D(θ ′ ),D(θ ′′ )}. Assume for definiteness that D(θ) = h+D(θ ′ ). Using the bound (3.2.4)
and the Markov property we get
E (9)
ε
[ [
Rh 4 E(9) R h
R −4
D(θ ′ )
]
[
E (9)
R h
R −4
D(θ ′′ )
]]
This completes the proof of the lemma.
[ ]]
≤ E ε
[E (9)
R 9 h
R −4
D(θ ′ )
[ ]
= E ε
(9) R −4
h+D(θ ′ )
[ ]
= E ε
(9) R −4
D(θ)
.
□
85

Q θ ε
As a consequence of Lemma 3.5.2, we get the bound
⎡
⎤
( (
⎣F −ε + w (a)) ) ∏ ( ) 4
∏ ( ) −4
V a V a
a∈L(θ)
a∈I(θ)
a∈L(θ)
⎦ ≤ E (9)
ε
[ ] ∏
R −4
D(θ)
[ ]
≤ E (9) ∏
0 R −4
D(θ)
= E(9) 0
[ ]
R
−4
1
D(θ) 2
a∈L(θ)½{δ≤d θ (0,a)≤M}
a∈L(θ)½{δ≤d θ (0,a)≤M}
∏
a∈L(θ)½{δ≤d θ (0,a)≤M}.
The last quantity is clearly integrable with respect to the measure Λ p (dθ). In addition, using
the continuity of F, it is easy to verify that
⎡
⎤
( ( ) ∏
lim
ε→0 Qθ ε
⎣F −ε + w (a)) ( ) 4
∏ ( ) −4
V a V a
⎦
a∈L(θ)
a∈I(θ) a∈L(θ)
⎡
⎤
( (
= Q θ 0
⎣F w (a)) ) ∏ ( ) 4
∏ ( ) −4
V a V a
⎦
a∈L(θ)
a∈I(θ)
a∈L(θ)
An application of the dominated convergence theorem now leads to (3.5.2).
Step 3. We now consider the left-hand side of formula (3.5.1). For every 0 < b < 1, we consider
the continuous function φ b : R + −→ [0,1] such that φ b (s) = 1 for every s ∈ [b,1/b], φ b (s) = 0
for every s ∈ R + \]b/2,2/b[, and φ b is linear on [b/2,b] and on [1/b,2/b]. From Theorem 3.3.1
and the definition of N 0 , we get
( ∫
lim
ε→0 ε−4 N 0 φ b (σ) ds 1 ... ds p F ( )
W s1 ,... ,W sp
)½{W>−ε}
]0,σ[ p ∫
= N 0
(φ b (σ) ds 1 ...ds p F ( ) )
(3.5.3)
W s1 ,...,W sp .
]0,σ[ p
Lemma 3.5.3. The following convergence holds :
∫
lim sup ε −4 N 0
((1 − φ b (σ)) ds 1 ...ds p F ( )
W s1 ,... ,W sp
)½{W>−ε} = 0.
b→0 ε∈(0,1)
]0,σ[ p
Proof : We first observe that
∫
ε −4 N 0
(½{σ−ε}
]0,σ[
(
p ) ∫ ∞
≤ b p ε −4 N 0 sup ζ s > δ , W > −ε = b p ε −4 dh
(3.5.4)
s∈[0,σ]
2h 2 Nh 0(W > −ε) ≤ b p C
6δ 3
δ
86

where the constant C is such that ε −4 N h 0 (W > −ε) ≤ Ch−2 , for every h > 0 and 0 < ε < 1 (cf
(3.4.12)). On the other hand, the Cauchy-Schwarz inequality gives
∫
ε −4 N 0
(½{σ>1/b} ds 1 ...ds p F ( )
W s1 ,... ,W sp
)½{W>−ε}
]0,σ[ p
⎛ ⎛( ∫
≤ ⎝ε −4 N 0
⎝ ds 1 ...ds p F ( ) ) ⎞
2½{W>−ε}⎞ 1/2
W s1 ,... ,W sp
⎠⎠
]0,σ[ p
×
(ε −4 N 0
(
σ > 1 b ,W > −ε )) 1/2
.
Note that we may write
⎛( ∫
N 0
⎝ ds 1 ...ds p F ( ) ) 2½{W>−ε}⎞
W s1 ,... ,W sp
⎠
]0,σ[ p
= N 0
( ∫
]0,σ[ 2p ds 1 ...ds 2p G ( W s1 ,...,W s2p
)½{W>−ε}
where G is a nonnegative symmetric function on W 2p , which is also bounded by 1. As a consequence
of (3.5.1) and Lemma 3.5.2, we then get
⎛( ∫
ε −4 N 0
⎝ ds 1 ...ds p F ( ) ) 2½{W>−ε}⎞
W s1 ,... ,W sp
⎠
]0,σ[ p
≤ (2p)!2 2p−1 E (9)
0
[ ] ∫ ∏
R
−4
1 Λ 2p (dθ)D(θ) −2 θ (∅,a)≤M} = C(p,δ,M) < ∞.
a∈L(θ)½{δ≤d
From Theorem 3.1.1 and a simple scaling argument, we have
(
ε −4 N 0 σ > 1 ) ∫ ∞
b , W > −ε = ε −4 ds
√
2πs
3 N(s) 0 (W > −ε) ≤ C′ b 1/2 .
By combining these estimates, we get
(3.5.5)
∫
ε −4 N 0
(½{σ>1/b} ds 1 ...ds p F ( )
W s1 ,... ,W sp
)½{W>−ε} ≤ (C ′ C(p,δ,M)) 1/2 b 1/4 .
]0,σ[ p
Lemma 3.5.3 follows from (3.5.4) and (3.5.5).
b −1
We can now complete the proof of Theorem 3.5.1. First, by monotone convergence,
( ∫
lim N 0 φ b (σ) ds 1 ...ds p F ( ) ) ( ∫
W s1 ,... ,W sp = N 0 ds 1 ...ds p F ( ) )
W s1 ,...,W sp .
b→0 ]0,σ[ p ]0,σ[ p
)
,
□
87

From (3.5.3) and Lemma 3.5.3, it then follows that
lim
ε→0 ε−4 N 0
( ∫
]0,σ[ p ds 1 ...ds p F ( W s1 ,... ,W sp
)½{W>−ε}
= N 0
( ∫
]0,σ[ p ds 1 ... ds p F ( W s1 ,... ,W sp
) ) .
)
Combining this with (3.5.1) and (3.5.2) gives Theorem 3.5.1.
□
Acknowledgement. The first author wishes to thank Philippe Chassaing for a stimulating
conversation which motivated the present work. We also thank the referee for several useful
remarks.
88

CHAPITRE 4
Asymptotics for rooted planar maps and scaling limits of
two-type spatial trees
4.1. Introduction
The main goal of the present work is to investigate asymptotic properties of large rooted
bipartite planar maps under the so-called Boltzmann distributions. This setting includes as a
special case the asymptotics as n → ∞ of the uniform distribution over rooted 2κ-angulations
with n faces, for any fixed integer κ ≥ 2. Boltzmann distributions over planar maps that are both
rooted and pointed have been considered recently by Marckert & Miermont [43] who discuss
in particular the profile of distances from the distinguished point in the map. Here we deal
with rooted maps and we investigate distances from the root vertex, so that our results do not
follow from the ones in [43], although many statements look similar. The specific results that
are obtained in the present work have found applications in the paper [40], which investigates
scaling limits of large planar maps.
Let us briefly discuss Boltzmann distributions over rooted bipartite planar maps. We consider
a sequence q = (q i ) i≥1 of weights (nonnegative real numbers) satisfying certain regularity
properties. Then, for each integer n ≥ 2, we choose a random rooted bipartite map M n with
n faces whose distribution is specified as follows : the probability that M n is equal to a given
bipartite planar map m is proportional to
n∏
i=1
q deg(fi )/2
where f 1 ,...,f n are the faces of m and deg(f i ) is the degree (that is the number of adjacent
edges) of the face f i . In particular we may take q κ = 1 and q i = 0 for i ≠ κ, and we get the
uniform distribution over rooted 2κ-angulations with n faces.
Theorem 4.2.5 below provides asymptotics for the radius and the profile of distances from
the root vertex in the random map M n when n → ∞. The limiting distributions are described
in terms of the one-dimensional Brownian snake driven by a normalized excursion. In particular
if R n denotes the radius of M n (that is the maximal distance from the root), then n −1/4 R n
converges to a multiple of the range of the Brownian snake. In the special case of quadrangulations
(q 2 = 1 and q i = 0 for i ≠ 2), these results were obtained earlier by Chassaing & Schaeffer
[14]. As was mentioned above, very similar results have been obtained by Marckert & Miermont
[43] in the setting of Boltzmann distributions over rooted pointed bipartite planar maps, but
considering distances from the distinguished point rather than from the root.
Similarly as in [14] or [43], bijection between trees and maps serve as a major tool in our
approach. In the case of quadrangulations, these bijections were studied by Cori & Vauquelin
[15] and then by Schaeffer [51]. They have been recently extended to bipartite planar maps by
Bouttier, di Francesco & Guitter [11]. More precisely, Bouttier, di Francesco & Guitter show that
89

ipartite planar maps are in one-to-one correspondence with well-labelled mobiles, where a welllabelled
mobile is a two-type spatial tree whose vertices are assigned positive labels satisfying
certain compatibility conditions (see section 4.2.4 for a precise definition). This bijection has the
nice feature that labels in the mobile correspond to distances from the root in the map. Then
the above mentioned asymptotics for random maps reduce to a limit theorem for well-labelled
mobiles, which is stated as Theorem 4.3.3 below. This statement can be viewed as a conditional
version of Theorem 11 in [43]. The fact that [43] deals with distances from the distinguished
point in the map (rather than from the root) makes it possible there to drop the positivity
constraint on labels. In the present work this constraint makes the proof significantly more
difficult. We rely on some ideas from Le Gall [39] who established a similar conditional theorem
for well-labelled trees. Although many arguments in Section 4.3 below are analogous to the ones
in [39], there are significant additional difficulties because we deal with two-type trees and we
condition on the number of vertices of type 1 rather than on the total number of vertices.
A key step in the proof of Theorem 4.3.3 consists in the derivation of estimates for the probability
that a two-type spatial tree remains on the positive half-line. As another application of
these estimates, we derive some information about separating vertices of uniform 2κ-angulations.
We show that with a probability close to one when n → ∞ a random rooted 2κ-angulation with
n faces will have a vertex whose removal disconnects the map into two components both having
size greater that n 1/2−ε . Related combinatorial results are obtained in [8]. More precisely, in
a variety of different models, Proposition 5 in [8] asserts that the second largest nonseparable
component of a random map of size n has size at most O(n 2/3 ). This suggests that n 1/2−ε in
our result could be replaced by n 2/3−ε .
The paper is organized as follows. In section 4.2, we recall some preliminary results and we
state our asymptotics for large random rooted planar maps. Section 4.3 is devoted to the proof of
Theorem 4.3.3 and to the derivation of Theorem 4.2.5 from asymptotics for well-labelled mobiles.
Finally Section 4.4 discusses the application to separating vertices of uniform 2κ-angulations.
4.2. Preliminaries
4.2.1. Boltzmann laws on planar maps. A planar map is a proper embedding, without
edge crossings, of a connected graph in the 2-dimensional sphere S 2 . Loops and multiple edges
are allowed. A planar map is said to be bipartite if all its faces have even degree. In this paper,
we will only be concerned with bipartite maps. The set of vertices will always be equipped with
the graph distance : if a and a ′ are two vertices, d(a,a ′ ) is the minimal number of edges on a
path from a to a ′ . If M is a planar map, we write F M for the set of its faces, and V M for the
set of its vertices.
A pointed planar map is a pair (M,τ) where M is a planar map and τ is a distinguished
vertex. Note that, since M is bipartite, if a and a ′ are two neighbouring vertices, then we have
|d(τ,a) − d(τ,a ′ )| = 1. A rooted planar map is a pair (M,⃗e ) where M is a planar map and ⃗e is
a distinguished oriented edge. The origin of ⃗e is called the root vertex. At last, a rooted pointed
planar map is a triple (M,e,τ) where (M,τ) is a pointed planar map and e is a distinguished
non-oriented edge. We can always orient e in such a way that its origin a and its end point
a ′ satisfy d(τ,a ′ ) = d(τ,a) + 1. Note that a rooted planar map can be interpreted as a rooted
pointed planar map by choosing the root vertex as the distinguished point.
Two pointed maps (resp. two rooted maps, two rooted pointed maps) are identified if there
exists an orientation-preserving homeomorphism of the sphere that sends the first map to the
second one and preserves the distinguished point (resp. the root edge, the distinguished point
90

and the root edge). Let us denote by M p (resp. M r , M r,p ) the set of all pointed bipartite maps
(resp. the set of all rooted bipartite maps, the set of all rooted pointed bipartite maps) up to
the preceding indentification.
Let us recall some definitions and propositions that can be found in [43]. Let q = (q i ,i ≥ 1)
be a sequence of nonnegative weights such that q i > 0 for at least one i > 1. For any planar
map M, we define W q (M) by
W q (M) = ∏
q deg(f)/2 ,
f∈F M
where we have written deg(f) for the degree of the face f. We require q to be admissible that is
Z q =
∑
W q (M) < ∞.
M∈M r,p
Note that the sum is over the set M r,p of all rooted pointed bipartite planar maps which is
countable thanks to the identification that was explained above. For k ≥ 1, we set N(k) = ( )
2k−1
k−1 .
For every weight sequence q, we define
f q (x) = ∑ k≥0
N(k + 1)q k+1 x k , x ≥ 0.
Let R q be the radius of convergence of the power series f q . Consider the equation
(4.2.1) f q (x) = 1 − x −1 , x > 0.
From Proposition 1 in [43], a sequence q is admissible if and only if equation (4.2.1) has at least
one solution, and then Z q is the solution of (4.2.1) that satisfies Z 2 q f ′ q (Z q) ≤ 1. An admissible
weight sequence q is said to be critical if it satisfies
(Z q ) 2 f ′ q (Z q ) = 1,
which means that the graphs of the functions x ↦−→ f q (x) and x ↦−→ 1 − 1/x are tangent at the
left of x = Z q . Furthermore, if Z q < R q , then q is said to be regular critical. This means that
the graphs are tangent both at the left and at the right of Z q . In what follows, we will only be
concerned with regular critical weight sequences.
Let q be a regular critical weight sequence. We define the Boltzmann distribution B r,p
q on the
set M r,p by
B r,p
q (M) = W q(M)
.
Z q
Let us now define Z (r)
q
by
Z (r)
q = ∑
q deg(f)/2 .
M∈M r f∈F M
Note that the sum is over the set M r of all rooted bipartite planar maps. From the fact that
Z q < ∞ it easily follows that Z q
(r) < ∞. We then define the Boltzmann distribution B r q on the
set M r by
B r q (M) = W q(M)
Z q
(r) .
∏
91

Let us turn to the special case of 2κ-angulations. A 2κ-angulation is a bipartite planar map
such that all faces have a degree equal to 2κ. If κ = 2, we recognize the well-known quadrangulations.
Let us set
(κ − 1)κ−1
α κ =
κ κ N(κ) .
We denote by q κ the weight sequence defined by q κ = α κ and q i = 0 for every i ∈ N \ {κ}. It is
proved in Section 1.5 of [43] that q κ is a regular critical weight sequence, and
Z qκ =
κ
κ − 1 .
For every n ≥ 1, we denote by U n κ (resp. U n κ ) the uniform distribution on the set of all rooted
pointed 2κ-angulations with n faces (resp. on the set of all rooted 2κ-angulations with n faces).
We have
B r,p
q κ
(· | #F M = n) = U n κ,
B r q κ
(· | #F M = n) = U n κ .
4.2.2. Two-type spatial Galton-Watson trees. We start with some formalism for discrete
trees. Set
U = ⋃ n≥0
N n ,
where N = {1,2,3,...} and by convention N 0 = {∅}. An element of U is a sequence u = u 1 ...u n ,
and we set |u| = n so that |u| represents the generation of u. In particular, |∅| = 0. If u = u 1 ...u n
and v = v 1 ...v m belong to U, we write uv = u 1 ...u n v 1 ...v m for the concatenation of u and
v. In particular, ∅u = u∅ = u. If v is of the form v = uj for u ∈ U and j ∈ N, we say that v is
a child of u, or that u is the father of v, and we write u = ˇv. More generally if v is of the form
v = uw for u,w ∈ U, we say that v is a descendant of u, or that u is an ancestor of v. The set U
comes with the natural lexicographical order such that u v if either u is an ancestor of v, or if
u = wa and v = wb with a ∈ U ∗ and b ∈ U ∗ satisfying a 1 < b 1 , where we have set U ∗ = U \ {∅}.
And we write u ≺ v if u v and u ≠ v.
A plane tree T is a finite subset of U such that
(i) ∅ ∈ T ,
(ii) u ∈ T \ {∅} ⇒ ǔ ∈ T ,
(iii) for every u ∈ T , there exists a number k u (T ) ≥ 0 such that uj ∈ T if and only if
1 ≤ j ≤ k u (T ).
We denote by T the set of all plane trees.
Let T be a plane tree and let ζ = #T − 1. The search-depth sequence of T is the sequence
u 0 ,u 1 ,...,u 2ζ of vertices of T wich is obtained by induction as follows. First u 0 = ∅, and then
for every i ∈ {0,1,... ,2ζ − 1}, u i+1 is either the first child of u i that has not yet appeared in
the sequence u 0 ,u 1 ,...,u i , or the father of u i if all children of u i already appear in the sequence
u 0 ,u 1 ,...,u i . It is easy to verify that u 2ζ = ∅ and that all vertices of T appear in the sequence
u 0 ,u 1 ,...,u 2ζ (of course some of them appear more that once). We can now define the contour
function of T . For every k ∈ {0,1,... ,2ζ}, we let C(k) denote the distance from the root of
the vertex u k . We extend the definition of C to the line interval [0,2ζ] by interpolating linearly
between successive integers. Clearly T is uniquely determined by its contour function C.
A discrete spatial tree is a pair (T ,U) where T ∈ T and U = (U v ,v ∈ T ) is a mapping from
the set T into R. If v is a vertex of T , we say that U v is the label of v. We denote by Ω the set of
92

all discrete spatial trees. If (T ,U) ∈ Ω we define the spatial contour function of (T ,U) as follows.
First if k is an integer, we put V (k) = U uk with the preceding notation. We then complete the
definition of V by interpolating linearly between successive integers. Clearly (T ,U) is uniquely
determined by the pair (C,V ).
Let (T ,U) ∈ Ω. We interpret (T ,U) as a two-type (spatial) tree by declaring that vertices of
even generations are of type 0 and vertices of odd generations are of type 1. We then set
T 0
T 1
= {u ∈ T : |u| is even},
= {u ∈ T : |u| is odd}.
Let us turn to random trees. We want to consider a particular family of two-type Galton-
Watson trees, in which vertices of type 0 only give birth to vertices of type 1 and vice-versa.
Let µ = (µ 0 ,µ 1 ) be a pair of offspring distributions, that is a pair of probability distributions
on Z + . If m 0 and m 1 are the respective means of µ 0 and µ 1 we assume that m 0 m 1 ≤ 1 and we
exclude the trivial case µ 0 = µ 1 = δ 1 where δ 1 stands for the Dirac mass at 1. We denote by P µ
the law of a two-type Galton-Watson tree with offspring distribution µ, meaning that for every
t ∈ T,
P µ (t) = ∏ u∈t 0 µ 0 (k u (t)) ∏ u∈t 1 µ 1 (k u (t)) ,
where t 0 (resp. t 1 ) is as above the set of all vertices of t with even (resp. odd) generation. The
fact that this formula defines a probability measure on T is justified in [43].
Let us now recall from [43] how one can couple plane trees with a spatial displacement in order
to turn them into random elements of Ω. To this end, let ν0 k,νk 1 be probability distributions on Rk
for every k ≥ 1. We set ν = (ν0 k,νk 1 ) k≥1. For every T ∈ T and x ∈ R, we denote by R ν,x (T ,dU)
the probability measure on R T which is characterized as follows. Let (Y u ,u ∈ T ) be a family
of independent random variables such that for u ∈ T with k u (T ) = k, Y u = (Y u1 ,... ,Y uk ) is
distributed according to ν0 k if u ∈ T 0 and according to ν1 k if u ∈ T 1 . We set X ∅ = x and for
every v ∈ T \ {∅},
X v = x + ∑
Y u ,
u∈]∅,v]
where ]∅,v] is the set of all ancestors of v distinct from the root ∅. Then R ν,x (T ,dU) is the law
of (X v ,v ∈ T ). We finally define for every x ∈ R a probability measure P µ,ν,x on Ω by setting
P µ,ν,x (dT dU) = P µ (dT )R ν,x (T ,dU).
4.2.3. The Brownian snake and the conditioned Brownian snake. Let x ∈ R. The
Brownian snake with initial point x is a pair (e,r x ), where e = (e(s),0 ≤ s ≤ 1) is a normalized
Brownian excursion and r x = (r x (s),0 ≤ s ≤ 1) is a real-valued process such that, conditionally
given e, r x is Gaussian with mean and covariance given by
• E[r x (s)] = x for every s ∈ [0,1],
• Cov(r x (s),r x (s ′ )) = inf
s≤t≤s ′ e(t) for every 0 ≤ s ≤ s′ ≤ 1.
We know from [37] that r x admits a continuous modification. From now on we consider only this
modification. In the terminology of [37] r x is the terminal point process of the one-dimensional
Brownian snake driven by the normalized Brownian excursion e and with initial point x.
93

Write P for the probability measure under which the collection (e,r x ) x∈R is defined. Note
that for every x > 0, we have ( )
P inf
s∈[0,1] rx (s) ≥ 0 > 0.
We may then define for every x > 0 a pair (e x ,r x ) which is distributed as the pair (e,r x ) under
the conditioning that inf s∈[0,1] r x (s) ≥ 0.
We equip C([0,1], R) 2 with the norm ‖(f,g)‖ = ‖f‖ u ∨ ‖g‖ u where ‖f‖ u stands for the
supremum norm of f. The following theorem is a consequence of Theorem 3.1.1.
Theorem 4.2.1. There exists a pair (e 0 ,r 0 ) such that (e x ,r x ) converges in distribution as
x ↓ 0 towards (e 0 ,r 0 ).
The pair (e 0 ,r 0 ) is the so-called conditioned Brownian snake with initial point 0.
Theorem 3.1.2 provides a useful construction of the conditioned object (e 0 ,r 0 ) from the
unconditioned one (e,r 0 ). In order to present this construction, first recall that there is a.s. a
unique s ∗ in (0,1) such that
r 0 (s ∗ ) = inf
s∈[0,1] r0 (s)
(see Lemma 16 in [45] or Proposition 3.2.5). For every s ∈ [0, ∞), write {s} for the fractional part
of s. According to Theorem 3.1.2, the conditioned snake (e 0 ,r 0 ) may be constructed explicitly
as follows : for every s ∈ [0,1],
e 0 (s) = e(s ∗ ) + e({s ∗ + s}) − 2
r 0 (s) = r 0 ({s ∗ + s}) − r 0 (s ∗ ).
inf e(t),
s∧{s ∗+s}≤t≤s∨{s ∗+s}
4.2.4. The Bouttier-di Francesco-Guitter bijection. We start with a definition. A
(rooted) mobile is a two-type spatial tree (T ,U) whose labels U v only take integer values and
such that the following properties hold :
(a) U v = Uˇv for every v ∈ T 1 .
(b) Let v ∈ T 1 such that k = k v (T ) ≥ 1. Let v (0) = ˇv be the father of v and let v (j) = vj
for every j ∈ {1,... ,k}. Then for every j ∈ {0,... ,k},
where by convention v (k+1) = v (0) .
U v(j+1) ≥ U v(j) − 1,
Furthermore, if U v ≥ 1 for every v ∈ T , then we say that (T ,U) is a well-labelled mobile.
Let T mob
1 denotes the set of all mobiles such that U ∅ = 1. We will now describe the Bouttierdi
Francesco-Guitter bijection from T mob
1 onto M r,p . This bijection can be found in section 2
in [11]. Note that [11] deals with pointed planar maps rather than with rooted pointed planar
maps. It is however easy to verify that the results described below are simple consequences of
[11].
Let (T ,U) ∈ T mob
1 . Recall that ζ = #T −1. Let u 0 ,u 1 ,... ,u 2ζ be the search-depth sequence of
T . It is immediate to see that u k ∈ T 0 if k is even and that u k ∈ T 1 if k is odd. The search-depth
sequence of T 0 is the sequence w 0 ,w 1 ,... ,w ζ defined by w k = u 2k for every k ∈ {0,1,... ,ζ}.
Notice that w 0 = w ζ = ∅. Although (T ,U) is not necesseraly well labelled, we may set for every
v ∈ T ,
U + v = U v − min{U w : w ∈ T } + 1,
94

and then (T ,U + ) is a well-labelled mobile. Notice that min{U + v : v ∈ T } = 1.
Suppose that the tree T is drawn in the plane and add an extra vertex ∂. We associate with
(T ,U + ) a bipartite planar map whose set of vertices is
T 0 ∪ {∂},
and whose edges are obtained by the following device : for every k ∈ {0,1,... ,ζ},
• if U + w k
= 1, draw an edge between w k and ∂ ;
• if U + w k
≥ 2, draw an edge between w k and the first vertex in the sequence w k+1 ,...,w ζ−1 ,
w 0 ,w 1 ,... ,w k−1 whose label is U + w k
− 1.
Notice that condition (b) in the definition of a mobile entails that U + w k+1
≥ U + w k
− 1 for every
k ∈ {0,1,... ,ζ − 1} and recall that min{U + w 0
,U + w 1
,... ,U + w ζ−1
} = 1. The preceding properties
ensure that whenever U + w k
≥ 2 there is at least one vertex among w k+1 ,... ,w ζ−1 ,w 0 ,... ,w k−1
with label U + w k
− 1. The construction can be made in such a way that edges do not intersect
(see section 2 in [11] for an example). The resulting planar graph is a bipartite planar map. We
view this map as a rooted pointed planar map by declaring that the distinguished vertex is ∂
and that the root edge is the one corresponding to k = 0 in the preceding construction.
It follows from [11] that the preceding construction yields a bijection Ψ r,p between T mob
1 and
M r,p . Furthermore it is not difficult to see that Ψ r,p satisfies the following two properties : let
(T ,U) ∈ T mob
1 and let M = Ψ r,p ((T ,U)),
(i) for every k ≥ 1, the set {f ∈ F M : deg(f) = 2k} is in one-to-one correspondence with
the set {v ∈ T 1 : k v (T ) = k − 1},
(ii) for every l ≥ 1, the set {a ∈ V M : d(∂,a) = l} is in one-to-one correspondence with the
set {v ∈ T 0 : U v − min{U w : w ∈ T } + 1 = l}.
We observe that if (T ,U) is a well-labelled mobile then U v
+ = U v for every v ∈ T . In
particular U ∅ + = 1. This implies that the root edge of the planar map Ψ r,p((T ,U)) contains
the distinguished point ∂. Then Ψ r,p ((T ,U)) can be identified to a rooted planar map, whose
root is an oriented edge between the root vertex ∂ and w 0 . Write T mob
1 for the set of all welllabelled
mobiles such that U ∅ = 1. Thus Ψ r,p induces a bijection Ψ r from the set T mob
1 onto
the set M r . Furthermore Ψ r satisfies the following two properties : let (T ,U) ∈ T mob
1 and let
M = Ψ r ((T ,U)),
(i) for every k ≥ 1, the set {f ∈ F M : deg(f) = 2k} is in one-to-one correspondence with
the set {v ∈ T 1 : k v (T ) = k − 1},
(ii) for every l ≥ 1, the set {a ∈ V M : d(∂,a) = l} is in one-to-one correspondence with the
set {v ∈ T 0 : U v = l}.
4.2.5. Boltzmann distribution on two-type spatial trees. Let q be a regular critical
weight sequence. We recall the following definitions from [43]. Let µ q 0 be the geometric
distribution with parameter f q (Z q ) that is
and let µ q 1
µ q 0 (k) = Z−1 q f q (Z q ) k , k ≥ 0,
be the probability measure defined by
µ q 1 (k) = Zk q N(k + 1)q k+1
, k ≥ 0.
f q (Z q )
95

From [43], we know that µ 1 has small exponential moments, and that the two-type Galton-
Watson tree associated with µ q = (µ q 0 ,µq 1 ) is critical.
Also, for every k ≥ 0, let ν0 k be the Dirac mass at 0 ∈ Rk and let ν1 k be the uniform distribution
on the set A k defined by
{
}
A k = (x 1 ,...,x k ) ∈ Z k : x 1 ≥ −1,x 2 − x 1 ≥ −1,... ,x k − x k−1 ≥ −1, −x k ≥ −1 .
We can say equivalently that ν1 k is the law of (X 1,...,X 1 + ... + X k ) where (X 1 ,... ,X k+1 ) is
uniformly distributed on the set B k defined by
{
}
B k = (x 1 ,...,x k+1 ) ∈ {−1,0,1,2,...} k+1 : x 1 + ... + x k+1 = 0 .
Notice that #A k = #B k = N(k+1). We set ν = ((ν k 0 ,νk 1 )) k≥1. The following result is Proposition
10 in [43]. However, we provide a short proof for the sake of completeness.
Proposition 4.2.2. The Boltzmann distribution B r,p
q is the image of the probability measure
P µ q ,ν,1 under the mapping Ψ r,p .
Proof : By construction, the probability measure P µ q ,ν,1 is supported on the set T mob
1 . Let
(Θ,u) ∈ T mob
1 . We have by the choice of ν,
P µ q ,ν,1 ((t,u)) = P µ q(t)R ν,1 (t, {u})
( ∏ ) −1Pµ
= N(k v (t) + 1) q(t).
v∈t 1
Now,
P µ q(t) =
∏ µ q 0 (k v(t)) ∏
µ q 1 (k v(t))
v∈t 0 v∈t 1
so that we arrive at
= ∏ (
Zq −1 f q(Z q ) kv(t)) ∏ Zq
kv(t) N(k v (t) + 1)q kv(t)+1
f q (Z q )
v∈t 0 v∈t 1
= Z −#t0
q
= Z −1
q
f q (Z q ) #t1 Z #t0 −1
q f q (Z q ) ∏ −#t1 N(k v (t) + 1) ∏
v∈t 1
∏
v∈t 1 q kv(t)+1
∏
v∈t 1 N(k v (t) + 1),
P µ q ,ν,1 ((t,u)) = Z −1
q
∏
v∈t 1 q kv(t)+1.
We set m = Ψ r,p ((t,u)). We have from the property (ii) satisfied by Ψ r,p ,
∏
Zq
−1 q kv(t)+1 = B r,p
q (m),
v∈t 1
which leads us to the desired result.
v∈t 1 q kv(t)+1
Let us introduce some notation. As µ q 0 (1) > 0, we have P µ(T 1 = n) > 0 for every n ≥ 1.
Then we may define, for every n ≥ 1 and x ∈ R,
P n µ q = P µ q (· | #T 1 = n ) ,
P n µ q ,ν,x = P µ q ,ν,x
(· | #T 1 = n ) .
□
96

Furthermore, we set for every (T ,U) ∈ Ω,
U = min { U v : v ∈ T 0 \ {∅} } ,
with the convention min ∅ = ∞. Finally we define for every n ≥ 1 and x ≥ 0,
P µ q ,ν,x = P µ q ,ν,x(· | U > 0),
P n µ q ,ν,x = P µ q ,ν,x
(· | #T 1 = n ) .
Corollary 4.2.3. The probability measure B r,p
q (· | #F M = n) is the image of P n µ q ,ν,1 under
the mapping Ψ r,p . The probability measure B r q is the image of P µ q ,ν,1 under the mapping Ψ r .
The probability measure B r q(· | #F M = n) is the image of P n µ q ,ν,1 under the mapping Ψ r .
Proof : The first assertion is a simple consequence of Proposition 4.2.2 together with the
property (i) satisfied by Ψ r,p . Recall from section 4.2.1 that we can identify the set M r to
a subset of M r,p in the following way. Let Υ : M r −→ M r,p be the mapping defined by
Υ((M,⃗e )) = (M,e,o) for every (M,⃗e ) ∈ M r , where o denotes the root vertex of the map
(M,⃗e ). We easily check that B r,p
q (· | M ∈ Υ(M r )) is the image of B r q under the mapping Υ.
This together with Proposition 4.2.2 yields the second assertion. The third assertion follows. □
At last, if q = q κ , we set µ κ 0 = µqκ 0 , µκ 1 = µqκ 1 and µ κ = (µ κ 0 ,µκ 1 ). We then verify that µκ 0 is
the geometric distribution with parameter 1/κ and that µ κ 1 is the Dirac mass at κ − 1. Recall
the notation U n κ and U n κ .
Corollary 4.2.4. The probability measure U n κ is the image of P n µ κ ,ν,1 under the mapping
Ψ r,p . The probability measure U n κ is the image of P n µ κ ,ν,1 under the mapping Ψ r.
4.2.6. Statement of the main result. We first need to introduce some notation. Let
M ∈ M r . We denote by o its root vertex. The radius R M is the maximal distance between o
and another vertex of M that is
R M = max{d(o,a) : a ∈ V M }.
The normalized profile of M is the probability measure λ M on {0,1,2,...} defined by
λ M (k) = #{a ∈ V M : d(o,a) = k}
, k ≥ 0.
#V M
Note that R M is the supremum of the support of λ M . It is also convenient to introduce the
rescaled profile. If M has n faces, this is the probability measure on R + defined by
λ (n)
M (A) = λ M
(
n 1/4 A
for any Borel subset A of R + . At last, if q is a regular critical weight sequence, we set
ρ q = 2 + Z 3 q f ′′
q (Z q).
Recall from section 4.2.3 that (e,r 0 ) denotes the Brownian snake with initial point 0.
Theorem 4.2.5. Let q be a regular critical weight sequence.
(i) The law of n −1/4 R M under the probability measure B r q(· | #F M = n) converges as
n → ∞ to the law of the random variable
( ) 1/4 (
)
4ρq
r 0 (s) − inf
9(Z q − 1)
0≤s≤1 r0 (s) .
sup
0≤s≤1
)
97

(ii) The law of the random measure λ (n)
M under the probability measure Br q (· | #F M = n)
converges as n → ∞ to the law of the random probability measure I defined by
∫ (
1
( ) 1/4 (
) ) 4ρq
〈I,g〉 = g
r 0 (t) − inf
9(Z q − 1)
0≤s≤1 r0 (s) dt.
0
(iii) The law of the rescaled distance n −1/4 d(o,a) where a is a vertex chosen uniformly at
random among all vertices of M, under the probability measure B r q (· | #F M = n)
converges as n → ∞ to the law of the random variable
( 4ρq
9(Z q − 1)
) 1/4 (
sup r 0 (s)
0≤s≤1
In the case q = q κ , the constant appearing in Theorem 4.2.5 is (4κ(κ − 1)/9) 1/4 . It is equal
to (8/9) 1/4 when κ = 2. The results stated in Theorem 4.2.5 in the special case q = q 2 were
obtained by Chassaing & Schaeffer [14] (see also Theorem 8.2 in [39]).
Obviously Theorem 4.2.5 is related to Theorem 3 proved by Marckert & Miermont [43]. Note
however that [43] deals with rooted pointed maps instead of rooted maps as we do and studies
distances from the distinguished point of the map rather than from the root vertex.
)
.
4.3. A conditional limit theorem for two-type spatial trees
Recall first some notation. Let q be a regular critical weight sequence, let µ q = (µ q 0 ,µq 1 ) be
the pair of offspring distributions associated with q and let ν = (ν k 0 ,νk 1 ) k≥1 be the family of
probability measures defined before Proposition 4.2.2.
If (T ,U) ∈ Ω, we denote by C its contour function and by V its spatial contour function.
Recall that C([0,1], R) 2 is equipped with the norm ‖(f,g)‖ = ‖f‖ u ∨ ‖g‖ u . The following
result is a special case of Theorem 11 in [43].
Theorem 4.3.1. Let q be a regular critical weight sequence. The law under P n µ q ,ν,0
⎛
of
(√ ) ( (9(Zq ) ) ⎞
ρq (Z
⎝ q − 1) C(2(#T − 1)t)
− 1) 1/4
V (2(#T − 1)t)
4 n 1/2 ,
⎠
4ρ q n 1/4
0≤t≤1
0≤t≤1
converges as n → ∞ to the law of (e,r 0 ). The convergence holds in the sense of weak convergence
of probability measures on C([0,1], R) 2 .
Note that Theorem 11 in [43] deals with the so-called height-process instead of the contour
process. However, we can deduce Theorem 4.3.1 from [43] by classical arguments (see e.g. [38]).
In this section, we will prove a conditional version of Theorem 4.3.1. Before stating this result,
we establish a corollary of Theorem 4.3.1. To this end we set
Q µ q = P µ q(· | k ∅ (T ) = 1),
Q µ q ,ν = P µ q ,ν,0(· | k ∅ (T ) = 1).
Notice that this conditioning makes sense since µ q 0 (1) > 0. We may also define for every n ≥ 1,
Q n µ q = Q µ
(· q | #T 1 = n ) ,
Q n µ q ,ν = Q µ q ,ν
(· | #T 1 = n ) .
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Corollary 4.3.2. Let q be a regular critical weight sequence. The law under Q n µ q ,ν of
⎛(√ ) ( (9(Zq ) ) ⎞
ρq (Z
⎝ q − 1) C(2(#T − 1)t)
− 1) 1/4
V (2(#T − 1)t)
4 n 1/2 ,
⎠
4ρ q n 1/4
0≤t≤1
0≤t≤1
converges as n → ∞ to the law of (e,r 0 ). The convergence holds in the sense of weak convergence
of probability measures on C([0,1], R) 2 .
Proof : We first introduce some notation. If (T ,U) ∈ Ω and w 0 ∈ T , we define a spatial
tree (T [w 0] ,U [w 0] ) by setting
T [w 0] = {v : w 0 v ∈ T },
and for every v ∈ T [w 0]
U [w 0]
v = U w0 v − U w0 .
Denote by C [w 0] the contour function and by V [w 0] the spatial contour function of (T [w 0] ,U [w 0] ).
As a consequence of Theorem 11 in [43], the law under Q n µ q ,ν
⎛
of
(√
ρq (Z
⎝ q − 1) C [1] ( 2 ( #T [1] − 1 ) t ) ) ( (9(Zq )
− 1) 1/4
V [1] ( 2 ( #T [1] − 1 ) t ) )
4
n 1/2 ,
4ρ q n 1/4
0≤t≤1
converges as n → ∞ to the law of (e,r 0 ). We then easily get the desired result.
0≤t≤1
Recall from section 4.2.3 that (e 0 ,r 0 ) denotes the conditioned Brownian snake with initial
point 0.
Theorem 4.3.3. Let q be a regular critical weight sequence. For every x ≥ 0, the law under
P n µ q ,ν,x
⎛
of
(√ ) ( (9(Zq ) ) ⎞
ρq (Z
⎝ q − 1) C(2(#T − 1)t)
− 1) 1/4
V (2(#T − 1)t)
4 n 1/2 ,
⎠
4ρ q n 1/4
0≤t≤1
0≤t≤1
converges as n → ∞ to the law of (e 0 ,r 0 ). The convergence holds in the sense of weak convergence
of probability measures on C([0,1], R) 2 .
To prove Theorem 4.3.3, we will follow the lines of the proof of Theorem 2.2 in [39]. From
now on, we set µ = µ q to simplify notation.
4.3.1. Rerooting two-type spatial trees. If T ∈ T, we say that a vertex v ∈ T is a leaf
of T if k v (T ) = 0 meaning that v has no child. We denote by ∂T the set of all leaves of T and
we write ∂ 0 T = ∂T ∩ T 0 for the set of leaves of T which are of type 0.
Let us recall some notation that can be found in section 3 in [39]. Recall that U ∗ = U \ {∅}.
If v 0 ∈ U ∗ and T ∈ T are such that v 0 ∈ T , we define k = k(v 0 , T ) and l = l(v 0 , T ) in the
following way. Write ζ = #T − 1 and u 0 ,u 1 ,...,u 2ζ for the search-depth sequence of T . Then
we set
k = min{i ∈ {0,1,... ,2ζ} : u i = v 0 },
l = max{i ∈ {0,1,... ,2ζ} : u i = v 0 },
⎞
⎠
□
99

which means that k is the time of the first visit of v 0 in the evolution of the contour of T and
that l is the time of the last visit of v 0 . Note that l ≥ k and that l = k if and only if v 0 ∈ ∂T .
For every t ∈ [0,2ζ − (l − k)], we set
Ĉ (v0) (t) = C(k) + C([[k − t]]) − 2 inf C(s),
s∈[k∧[k−t],k∨[k−t]]
where C is the contour function of T and [[k−t]] stands for the unique element of [0,2ζ) such that
[[k−t]]−(k−t) = 0 or 2ζ. Then there exists a unique plane tree ̂T (v 0) ∈ T whose contour function
is Ĉ(v 0) . Informally, ̂T (v 0) is obtained from T by removing all vertices that are descendants of
v 0 and by re-rooting the resulting tree at v 0 . Furthermore, if v 0 = u 1 ...u n , then we see that
̂v 0 = 1u n ...u 2 belongs to ̂T (v 0) . In fact, ̂v 0 is the vertex of ̂T (v 0) corresponding to the root of
the initial tree. At last notice that k ∅ (̂T (v 0) ) = 1.
If T ∈ T and w 0 ∈ T , we set
T (w 0) = T \ {w 0 u ∈ T : u ∈ U ∗ }.
The following lemma is an analogue of Lemma 3.1 in [39] for two-type Galton-Watson trees.
Note that in what follows, two-type trees will always be re-rooted at a vertex of type 0.
Recall the definition of the probability measure Q µ .
Lemma 4.3.4. Let v 0 ∈ U ∗ be of the form v 0 = 1u 2 ... u 2p for some p ∈ N. Assume that
Q µ (v 0 ∈ T ) > 0. Then the law of the re-rooted tree ̂T (v 0) under Q µ (· | v 0 ∈ T ) coincides with
the law of the tree T (bv 0) under Q µ (· | ̂v 0 ∈ T ).
Proof : We first notice that
Q µ (v 0 ∈ T ) = µ 1
({
u 2 ,u 2 + 1,... }) µ 0
({
u 3 ,u 3 + 1,... }) ... µ 1
({
u 2p ,u 2p + 1,... }) ,
so that
Q µ (v 0 ∈ T ) = Q µ (̂v 0 ∈ T ) > 0.
In particular, both conditionings of Lemma 4.3.4 make sense. Let t be a two-type tree such that
̂v 0 ∈ ∂ 0 t and k ∅ (t) = 1. Since the trees t and ̂t (bv 0) represent the same graph, we have
) ∏
Q µ
(T (bv 0) = t = µ 0 (k u (t)) ∏ µ 1 (k u (t))
u∈t 1
=
=
which implies the desired result.
u∈t 0 \{∅,bv 0 }
∏
u∈t 0 \{∅,bv 0 }
∏
u∈bt (bv 0 ),0 \{∅,v 0 }
(
= Q µ T (v0) = ̂t )
(bv 0)
)
(v
= Q µ
(̂T 0 ) = t ,
µ 0 (deg(u) − 1) ∏ u∈t 1 µ 1 (deg(u) − 1)
µ 0 (deg(u) − 1)
∏
u∈bt (bv 0 ),1 µ 1 (deg(u) − 1)
Before stating a spatial version of Lemma 4.3.4, we establish a symmetry property of the
collection of measures ν. To this end, we let ˜ν 1 k be the image measure of νk 1 under the mapping
(x 1 ,... ,x k ) ∈ R k ↦−→ (x k ,...,x 1 ) and we set ˜ν = ((ν0 k, ˜νk 1 )) k≥1.
□
100

Lemma 4.3.5. For every k ≥ 1 and every j ∈ {1,... ,k}, the measures ν1 k and ˜νk 1 are invariant
under the mapping φ j : R k −→ R k defined by
φ j (x 1 ,...,x k ) = (x j+1 − x j ,... ,x k − x j , −x j ,x 1 − x j ,...,x j−1 − x j ).
Proof : Recall the definition of the sets A k and B k . Let ρ k be the uniform distribution on
B k . Then ν k 1 is the image measure of ρk under the mapping ϕ k : B k −→ A k defined by
For every (x 1 ,...,x k+1 ) ∈ R k+1 we set
ϕ k (x 1 ,... ,x k+1 ) = (x 1 ,x 1 + x 2 ,... ,x 1 + ... + x k ).
p j (x 1 ,... ,x k+1 ) = (x j+1 ,...,x k+1 ,x 1 ,... ,x j ).
It is immediate that ρ k is invariant under the mapping p j . Furthermore φ j ◦ ϕ k (x) = ϕ k ◦ p j (x)
for every x ∈ B k , which implies that ν k 1 is invariant under φ j.
At last for every (x 1 ,... ,x k ) ∈ R k we set
S(x 1 ,...,x k ) = (x k ,...,x 1 ).
Then φ j ◦ S = S ◦ φ k−j+1 , which implies that ˜ν k 1 is invariant under φ j.
If (T ,U) ∈ Ω and v 0 ∈ T 0 , the re-rooted spatial tree (̂T (v0),Û(v0) ) is defined as follows. For
every vertex v ∈ ̂T (v0),0 , we set
Û (v 0)
v = U v − U v0 ,
where v is the vertex of the initial tree T corresponding to v, and for every vertex v ∈ ̂T (v0),1 ,
we set
Û (v 0)
v = Û(v 0)
ˇv .
Note that, since v 0 ∈ T 0 , v ∈ ̂T (v0) is of type 0 if and only if v ∈ T is of type 0.
If (T ,U) ∈ Ω and w 0 ∈ T , we also consider the spatial tree (T (w 0) ,U (w 0) ) where U (w 0) is the
restriction of U to the tree T (w 0) .
Recall the definition of the probability measure Q µ,ν .
Lemma 4.3.6. Let v 0 ∈ U ∗ be of the form v 0 = 1u 2 ... u 2p for some p ∈ N. Assume that
Q µ (v 0 ∈ T ) > 0. Then the law of the re-rooted spatial tree (̂T (v 0) ,Û(v 0) ) under Q µ,eν (· | v 0 ∈ T )
coincides with the law of the spatial tree (T (bv 0) ,U (bv 0) ) under Q µ,ν (· | ̂v 0 ∈ T ).
Lemma 4.3.6 is a consequence of Lemma 4.3.4 and Lemma 4.3.5. We leave details to the
reader.
If (T ,U) ∈ Ω, we denote by ∆ 0 = ∆ 0 (T ,U) the set of all vertices of type 0 with minimal
spatial position :
∆ 0 = { v ∈ T 0 : U v = min{U w : w ∈ T } } .
We also denote by v m the first element of ∆ 0 in the lexicographical order. The following two
lemmas can be proved from Lemma 4.3.6 in the same way as Lemma 3.3 and Lemma 3.4 in [39].
Lemma 4.3.7. For any nonnegative measurable functional F on Ω,
(
(v
Q µ,eν
(F
(̂T m) ,Û(vm))½{#∆ 0 =1,v m∈∂ 0 T })
= Q µ,ν F(T ,U)(#∂0 T )½{U>0})
.
Lemma 4.3.8. For any nonnegative measurable functional F on Ω,
⎛
Q µ,eν
⎝
∑ (v
F
(̂T )⎞ 0 ) ( ) ,Û(v 0) ⎠ = Q µ,ν F(T ,U)(#∂0 T )½{U≥0} .
v 0 ∈∆ 0 ∩∂ 0 T
101
□

4.3.2. Estimates for the probability of staying on the positive side. In this section
we will derive upper and lower bounds for the probability P n µ,ν,x(U > 0) as n → ∞. We first
state a lemma which is a direct consequence of Lemma 17 in [43].
Lemma 4.3.9. There exist constants c 0 > 0 and c 1 > 0 such that
n 3/2 (
P µ #T 1 = n )
−→ c 0 ,
n→∞
n 3/2 (
Q µ #T 1 = n )
−→ c 1 .
n→∞
We now establish a preliminary estimate concerning the number of leaves of type 0 in a tree
with n vertices of type 1.
Lemma 4.3.10. There exists a constant β 0 > 0 such that for every n sufficiently large,
)
P µ
(|(#∂ 0 T ) − m 1 µ 0 (0)n| > n 3/4 ,#T 1 = n ≤ e −nβ0 .
Proof : Let T be a two-type tree. Recall that ζ = #T − 1. Let
v(0) = ∅ ≺ v(1) ≺ ... ≺ v(ζ)
be the vertices of T listed in lexicographical order. For every n ∈ {0,1,... ,ζ} we define R n =
(R n (k)) k≥1 as follows. For every k ∈ {1,... , |v(n)|}, R n (k) is the number of younger brothers of
the ancestor of v(n) at generation k. Here younger brothers are those brothers which have not
yet been visited at time n in search-depth sequence. For every k > |v(n)|, we set R n (k) = 0.
Standard arguments (see e.g. [41] for similar results) show that (R n , |v(n)|) 0≤n≤ζ has the same
distribution as (R n ′ ,h′ n ) 0≤n≤T ′ −1, where (R n ′ ,h′ n ) n≥0 is a Markov chain whose transition kernel
is given by :
(
)
• S ((r 1 ,...,r h ,0,...),h),((r 1 ,...,r h ,k − 1,0,...),h + 1) = µ i (k) for k ≥ 1, h ≥ 0 and
r 1 ,...,r h ≥ 0,
(
)
• S ((r 1 ,...,r h ,0,...),h),((r 1 ,...,r l − 1,0,...),l) = µ i (0), where
l = inf{m ≥ 1 : r m > 0},
for h ≥ 1 and r 1 ,...,r h ≥ 0 such that {m ≥ 1 : r m > 0} ≠ ∅,
• S((0,h),(0,0)) = µ i (0) for every h ≥ 0,
where i = 0 if h is even, and i = 1 if h is odd, and finally
T ′ = inf { n ≥ 1 : (R ′ n,h ′ n) = (0,0) } .
Write P ′ for the probability measure under which (R n ′ ,h′ n ) n≥0 is defined. We define a sequence
of stopping times (τ j ′) j≥0 by τ 0 ′ = inf{n ≥ 0 : h′ n is odd} and τ j+1 ′ = inf{n > τ j ′ : h′ n is odd} for
every j ≥ 0. At last we set for every j ≥ 0,
} ( ( ))
X j =½{
′ h ′ τ j ′ +1 = h′ τ j ′ + 1 1 + R
τ ′ j ′ +1 h ′ τ j ′ +1 .
Since #T 0 = 1 + ∑ u∈T 1 k u(T ), we have
⎛
)
P µ
(|#T 0 − m 1 n| > n 3/4 ,#T 1 = n = P ′ ⎝ ∣ ⎞
n−1
∑
∣ X j ′ − m 1n + 1∣ > n 3/4 ,τ n−1 ′ < T ′ < τ n
′ ⎠
≤
j=0
⎛
P ′ ⎝ ∣ ⎞
n−1
∑
∣ X j ′ − m 1 n∣ > n 3/4 − 1⎠ .
102
j=0

Thanks to the strong Markov property, the random variables X j ′ are independent and distributed
according to µ 1 . A standard moderate deviations inequality ensures the existence of a positive
constant β 1 > 0 such that for every n sufficiently large,
( ∣∣#T
(4.3.1) P 0 µ − m 1 n ∣ )
> n 3/4 ,#T 1 = n ≤ e −nβ1 .
In the same way as previously, we define another sequence of stopping times (θ j ′) j≥0 by θ 0 ′ = 0
and θ j+1 ′ = inf{n > θ′ j : h′ n is even} for every j ≥ 0 and we set for every j ≥ 0,
}
Y j ′ =½{h ′ θ j ′ +1 ≤ h′ θ j
′ .
Using the sequences (θ j ′) j≥0 and (Y j ′)
j≥0, an argument similar to the proof of (4.3.1) shows that
there exists a positive constant β 2 > 0 such that for every n sufficiently large,
)
(4.3.2) P µ
(|#(∂ 0 T ) − µ 0 (0)n| > n 5/8 ,#T 0 = n ≤ e −nβ2 .
From (4.3.1), we get for n sufficiently large,
)
P µ
(|(#∂ 0 T ) − m 1 µ 0 (0)n| > n 3/4 ,#T 1 = n
≤ e −nβ1 + P µ
(|(#∂ 0 T ) − m 1 µ 0 (0)n| > n 3/4 , |#T 0 − m 1 n| ≤ n 3/4) .
However, for n sufficiently large,
P µ
(
|(#∂ 0 T ) − m 1 µ 0 (0)n| > n 3/4 , |#T 0 − m 1 n| ≤ n 3/4)
=
≤
≤
⌊n 3/4 +m 1 n⌋
∑
k=⌈−n 3/4 +m 1 n⌉
⌊n 3/4 +m 1 n⌋
∑
k=⌈−n 3/4 +m 1 n⌉
⌊n 3/4 +m 1 n⌋
∑
k=⌈−n 3/4 +m 1 n⌉
)
P µ
(|(#∂ 0 T ) − m 1 µ 0 (0)n| > n 3/4 ,#T 0 = k
)
P µ
(|(#∂ 0 T ) − µ 0 (0)k| > (1 − µ 0 (0))n 3/4 ,#T 0 = k
)
P µ
(|(#∂ 0 T ) − µ 0 (0)k| > k 5/8 ,#T 0 = k .
At last, we use (4.3.2) to obtain for n sufficiently large,
P µ
(|(#∂ 0 T ) − m 1 µ 0 (0)n| > n 3/4 , |#T 0 − m 1 n| ≤ n 3/4) ≤ (2n 3/4 + 1)e −Cnβ2 ,
where C is a positive constant. The desired result follows by combining this last estimate with
(4.3.1). □
We will now state a lemma which plays a crucial role in the proof of the main result of this
section. To this end, recall the definition of v m and set for every n ≥ 1,
Q n µ = Q µ
(· | #T 1 = n ) ,
Q n µ,ν = Q µ,ν
(· | #T 1 = n ) .
Lemma 4.3.11. There exists a constant c > 0 such that for every n sufficiently large,
Q n µ,ν (v m ∈ ∂ 0 T ) ≥ c.
103

Proof : The proof of this lemma is similar to the proof of Lemma 4.3 in [39]. Nevertheless,
we give a few details to explain how this proof can be adapted to our context.
Choose p ≥ 1 such that µ 1 (p) > 0. Under Q µ,ν (· | k 1 (T ) = p, k 11 (T ) = ... = k 1p (T ) = 2),
we can define 2p spatial trees {(T ij ,U ij ),i = 1,... ,p,j = 1,2} as follows. For i ∈ {1,... ,p} and
j = 1,2, we set
T ij = {∅} ∪ {1v : 1ijv ∈ T },
U ij
∅
= 0 and Uij 1v = U 1ijv − U 1i if 1ijv ∈ T . Then under the probability measure Q µ,ν (· |
k 1 (T ) = p, k 11 (T ) = ... = k 1p (T ) = 2), the trees {(T ij ,U ij ),i = 1,... ,p,j = 1,2} are
independent and distributed according to Q µ,ν . Furthermore, we notice that under the measure
Q µ,ν (· | k 1 (T ) = p, k 11 (T ) = ... = k 1p (T ) = 2), we have with an obvious notation
( {#T 11,1 + #T 12,1 = n − 2p + 1 } ∩ {U 11 < 0} ∩ { v 11
m ∈ ∂ 0 T 11} ∩
∩ { U 12 ≥ 0 } ∩
⋂
2≤i≤p,j=1,2
So we have for n ≥ 1 + 2p,
Q µ,ν
(
#T 1 = n,v m ∈ ∂ 0 T )
(4.3.3)
⋂
2≤i≤p
{U 1i ≥ 0}
{
T ij = {∅,1,11,... ,1p}, U ij ≥ 0 } ) ⊂ { #T 1 = n,v m ∈ ∂ 0 T } .
n−2p
∑ (
≥ C(µ,ν,p) Q µ,ν #T 1 = j,v m ∈ ∂ 0 T ) (
Q µ,ν #T 1 = n − j − 2p + 1, U ≥ 0 ) ,
j=1
where
C(µ,ν,p) = µ 1 (p) 2p−1 µ 0 (2) p µ 0 (0) 2p(p−1) ν p (
1 (−∞,0) × [0,+∞)
p−1 ) ν p 1 ([0,+∞)p ) 2(p−1) .
From (4.3.3), we are now able to get the result by following the lines of the proof of Lemma 4.3
in [39].
□
We can now state the main result of this section.
Proposition 4.3.12. Let K > 0. There exist constants γ 1 > 0, γ 2 > 0, ˜γ 1 > 0 and ˜γ 2 > 0
such that for every n sufficiently large and for every x ∈ [0,K],
˜γ 1
n ≤ Qn µ,ν(U > 0) ≤ ˜γ 2
n ,
γ 1
n ≤ Pn µ,ν,x (U > 0) ≤ γ 2
n .
Proof : The proof of Proposition 4.3.12 is similar to the proof of Proposition 4.2 in [39].
The major difference comes from the fact that we cannot easily get an upper bound for #(∂ 0 T )
on the event {#T 1 = n}. In what follows, we will explain how to circumvent this difficulty.
We first use Lemma 4.3.7 with F(T ,U) =½{T 1 =n}. Since #̂T (v 0),1 = #T 1 if v 0 ∈ ∂ 0 T , we
get
(4.3.4) Q µ,ν
(
#(∂0 T )½{#T 1 =n, U>0})
≤ Qµ
(
#T 1 = n ) .
On the other hand, we have
( )
Q µ,ν #(∂0 T )½{#T 1 =n, U>0}
≥ m 1 µ 0 (0)nQ µ,ν
(
#T 1 = n, U > 0 ) − Q µ,ν
(
|#(∂0 T ) − m 1 µ 0 (0)n|½{#T 1 =n, U>0})
.
(4.3.5)
104

Now thanks to Lemma 4.3.10, we have for n sufficiently large,
( )
Q µ,ν |#(∂0 T ) − m 1 µ 0 (0)n|½{#T 1 =n, U>0}
≤ n 3/4 (
Q µ,ν #T 1 = n, U > 0 ) ( )
+ Q µ,ν #(∂0 T )½{#T 1 =n, U>0}
(4.3.6)
+ m 1 µ 0 (0)nQ µ,ν (|#(∂ 0 T ) − m 1 µ 0 (0)n| > n 3/4 ,#T 1 = n)
≤ n 3/4 Q µ,ν
(
#T 1 = n, U > 0 ) + Q µ,ν
(
#(∂0 T )½{#T 1 =n, U>0})
+ m1 µ 0 (0)ne −nβ0 .
From (4.3.4), (4.3.5) and (4.3.6) we get for n sufficiently large
(m 1 µ 0 (0)n − n 3/4 )Q µ,ν
(
#T 1 = n, U > 0 ) ≤ 2Q µ
(
#T 1 = n ) + m 1 µ 0 (0)ne −nβ0 .
Using Lemma 4.3.9 it follows that
lim supnQ n µ,ν (U > 0) ≤ 2
n→∞
m 1 µ 0 (0) ,
which ensures the existence of ˜γ 2 .
Let us now use Lemma 4.3.8 with
F(T ,U) =½{T 1 =n,#(∂ 0 T )≤m 1 µ 0 (0)n+n 3/4 } .
Since #(∂ 0 T ) = #(∂ 0 ̂T
(v 0 ) ) if v 0 ∈ ∂ 0 T , we have for n sufficiently large,
)
Q µ,ν
(#(∂ 0 T )½{#T 1 =n,#(∂ 0 T )≤m 1 µ 0 (0)+n 3/4 , U≥0}
)
= Q µ,eν
(#(∆ 0 ∩ ∂ 0 T )½{#T 1 =n,#(∂ 0 T )≤m 1 µ 0 (0)+n 3/4 }
)
= Q µ,ν
(#(∆ 0 ∩ ∂ 0 T )½{#T 1 =n,#(∂ 0 T )≤m 1 µ 0 (0)+n 3/4 }
(
≥ Q µ,ν #(∆ 0 ∩ ∂ 0 T ) ≥ 1, #T 1 = n, #(∂ 0 T ) ≤ m 1 µ 0 (0) + n 3/4)
(
≥ Q µ,ν v m ∈ ∂ 0 T , #T 1 = n, #(∂ 0 T ) ≤ m 1 µ 0 (0) + n 3/4)
(
≥ Q µ,ν vm ∈ ∂ 0 T , #T 1 = n ) )
− Q µ,ν
(#(∂ 0 T ) > m 1 µ 0 (0) + n 3/4 ,#T 1 = n
(4.3.7)
≥ c Q µ,ν
(
#T 1 = n ) − e −nβ0 ,
where the last inequality comes from Lemma 4.3.10 and Lemma 4.3.11. On the other hand,
)
Q µ,ν
(#(∂ 0 T )½{#T 1 =n,#(∂ 0 T )≤m 1 µ 0 (0)+n 3/4 , U≥0}
(
≤ m 1 µ 0 (0)n + n 3/4) (
(4.3.8)
Q µ,ν #T 1 = n, U ≥ 0 ) .
Then (4.3.7), (4.3.8) and Lemma 4.3.9 imply that for n sufficiently large,
c
(4.3.9) lim inf
n→∞ nQn µ,ν(U ≥ 0) ≥
m 1 µ 0 (0) .
105

Recall that p ≥ 1 is such that µ 1 (p) > 0. Also recall the definition of the spatial tree (T [w 0] ,U [w 0] ).
From the proof of Corallary 4.3.2, we have
(
P µ,ν,0 U > 0,#T 1 = n ) ≥ P µ,ν,0
(k ∅ (T ) = 1,k 1 (T ) = p,U 11 > 0,... ,U 1p > 0,
)
#T [11],1 = n − 1, U [11] ≥ 0, T [12] = ... = T [1p] = {∅}
(
≥ C 2 (µ,ν,p)P µ,ν,0 U ≥ 0,#T 1 = n − 1 )
(4.3.10)
≥ µ 0 (1)C 2 (µ,ν,p)Q µ,ν
(
U ≥ 0,#T 1 = n − 1 ) ,
where we have set
C 2 (µ,ν,p) = µ 0 (1)µ 1 (p)µ 0 (0) p−1 ν p 1 ((0,+∞)p ) .
We then deduce the existence of γ 1 from (4.3.9), (4.3.10) and Lemma 4.3.9. A similar argument
gives the existence of ˜γ 1 .
At last, we define m ∈ N by the condition (m − 1)p < K ≤ mp. For every l ∈ N, we define
1 l ∈ U by 1 l = 11... 1, |1 l | = l. Notice that ν p 1 ({(p,p − 1,... ,1)}) = N(p + 1)−1 . By arguing on
the event
{k ∅ (T ) = k 11 (T ) = ... = k 1 2m−2 = 1,k 1 (T ) = ... = k 1 2m−1(T ) = p},
we see that for every n ≥ m,
(4.3.11) Q µ,ν
(
#T 1 = n, U > 0 ) ≥ C 3 (µ,ν,p,m)P µ,ν,K
(
#T 1 = n − m, U > 0 ) ,
with
C 3 (µ,ν,p,m) = µ 0 (1) m−1 µ 1 (p) m µ 0 (0) m(p−1) N(p + 1) −m .
Thanks to Lemma 4.3.9, (4.3.11) yields for every n sufficiently large,
which gives the existence of γ 2 .
P n µ,ν,K (U > 0) ≤ 2c 1
c 0
˜γ 2
C 3 (µ,ν,p,m)
4.3.3. Asymptotic properties of conditioned trees. We first introduce a specific notation
for rescaled contour and spatial contour processes. For every n ≥ 1 and every t ∈ [0,1],
we set
√
C (n) ρq (Z q − 1) C(2(#T − 1)t)
(t) =
4 n 1/2 ,
( )
V (n) 9(Zq − 1) 1/4
V (2(#T − 1)t)
(t) =
4ρ q n 1/4 .
In this section, we will get some information about asymptotic properties of the pair (C (n) ,V (n) )
under P n µ,ν,x. We will consider the conditioned measure
Q n µ,ν = Qn µ,ν (· | U > 0).
Berfore stating the main result of this section, we will establish three lemmas. The first one is
the analogue of Lemma 6.2 in [39] for two-type spatial trees and can be proved in a very similar
way.
1
n ,
□
106

Lemma 4.3.13. There exists a constant c > 0 such that, for every measurable function F on
Ω with 0 ≤ F ≤ 1,
(
( )
Qµ,ν(F(T n ,U)) ≤ c Q n (v
µ,eν F
(̂T m) ,Û(vm))) + O n 5/2 e −nβ 0
,
( )
where the constant β 0 is defined in Lemma 4.3.10 and the estimate O n 5/2 e −nβ 0
for the remainder
holds uniformly in F.
Recall the notation ˇv for the “father” of the vertex v ∈ T \ {∅}.
Lemma 4.3.14. For every ε > 0,
(
(4.3.12) Q n µ,ν
Likewise, for every ε > 0 and x ≥ 0,
(
(4.3.13) P n µ,ν,x
sup
v∈T \{∅}
sup
v∈T \{∅}
|U v − Uˇv |
n 1/4
|U v − Uˇv |
n 1/4
> ε
> ε
)
)
−→ 0.
n→∞
−→ 0.
n→∞
Proof : Let ε > 0. First notice that the probability measure ν1 k is supported on the set
{−k, −k + 1,... ,k} k . Then we have Q n µ,ν a.s. or P n µ,ν,x a.s.,
sup |U v − Uˇv | = sup |U v − Uˇv | ≤ sup k v (T ).
v∈T \{∅}
v∈T 0 \{∅}
v∈T 1
Now, from Lemma 16 in [43], there exists a constant α 0 > 0 such that for all n sufficiently large,
(
)
Q µ sup k v (T ) > εn 1/4 ≤ e −nα0 ,
v∈T
(
1 )
P µ sup k v (T ) > εn 1/4 ≤ e −nα0 .
v∈T 1
Our assertions (4.3.12) and (4.3.13) easily follow using also Lemma 4.3.9 and Proposition
4.3.12. □
Recall the definition of the re-rooted tree (̂T (v 0) ,Û(v 0) ). Its contour and spatial contour functions
(Ĉ(v 0) , ̂V (v 0) ) are defined on the line interval [0,2(#̂T (v 0) − 1)]. We extend these functions
to the line interval [0,2(#T − 1)] by setting Ĉ(v 0) (t) = 0 and ̂V (v 0) (t) = 0 for every
t ∈ [2(#̂T (v 0) − 1),2(#T − 1)]. Also recall the definition of v m .
At last, recall that (e 0 ,r 0 ) denotes the conditioned Brownian snake.
Lemma 4.3.15. The law under Q n µ,ν
⎛
of
(√ )
ρq (Z
⎝ q − 1) Ĉ (vm) (2(#T − 1)t)
4 n 1/2
0≤t≤1
,
( (9(Zq
− 1)
4ρ q
) )
1/4 ̂V
(v m) (2(#T − 1)t)
n 1/4
0≤t≤1
converges to the law of (e 0 ,r 0 ). The convergence holds in the sense of weak convergence of
probability measures on the space C([0,1], R) 2 .
Proof : From Corollary 4.3.2 and the Skorokhod representation theorem, we can construct
on a suitable probability space a sequence a sequence (T n ,U n ) and a Brownian snake (,Ö0 ),
⎞
⎠
107

such that each pair (T n ,U n ) is distributed according to Q n µ,ν, and such that if we write (C n ,V n )
for the contour and spatial contour functions of (T n ,U n ) and ζ n = #T n − 1, we have
⎛(√ ) ( (9(Zq ) ) ⎞
ρq (Z
(4.3.14) ⎝ q − 1) C n (2ζ n t)
− 1) 1/4
V n (2ζ n t)
4 n 1/2 ,
⎠
4ρ q n 1/4 −→ (,Ö0 ),
n→∞
uniformly on [0,1], a.s.
0≤t≤1
0≤t≤1
Then if (T ,U) ∈ Ω and v 0 ∈ T , we introduce a new spatial tree (̂T (v 0) ,Ũ(v 0) ) by setting for
every w ∈ ̂T (v 0)
Ũ (v 0)
w = U w − U v0 ,
where w is the vertex corresponding to w in the initial tree (in contrast with the definition of
Û (v 0)
w , Ũ(v 0)
v does not necesseraly coincide with Ũ(v 0)
ˇv when v is of type 1). We denote by Ṽ (v0) the
spatial contour function of (̂T (v0),Ũ(v0) ), and we set Ṽ (v0) (t) = 0 for t ∈ [2(#̂T (v0) − 1),2(#T −
1)]. Note that, if w ∈ ̂T (v0) is either a vertex of type 0 or a vertex of type 1 which does not
belong to the ancestral line of ̂v 0 , then
Û (v 0)
w = Ũ(v 0)
w ,
whereas if w ∈ ̂T (v 0),1 belongs to the ancestral line of ̂v 0 , then
Û (v 0)
w = Ũ(v 0)
ˇw .
Then we have
∣ ∣∣ (4.3.15) sup Û w − Ũw∣ ≤
w∈ bT (v 0 )
sup
w∈T \{∅}
|U w − U ˇw |.
Write vm n for the first vertex realizing the minimal spatial position in T n. In the same way as
in the derivation of (18) in the proof of Proposition 6.1 in [39], it follows from (4.3.14) that
⎛(√ ) (
ρq (Z
⎝ q − 1) Ĉ (vn m
n )
(9(Zq ) ) ⎞
(2ζ n t)
− 1) 1/4 (v Ṽ
m n n ) (2ζ n t)
4 n 1/2 ,
⎠
4ρ q n 1/4 −→ (0 ,Ö0 ),
n→∞
0≤t≤1
0≤t≤1
uniformly on [0,1], a.s., where the conditioned pair (0,Ö0 ) is constructed from the unconditioned
one (,Ö0 ) as explained in section 4.2.3. Let ε > 0. We deduce from (4.3.15) that
(
∣ ) (
)
P ′ ̂V (vn m)
n (2ζ n t)
sup
∣ n 1/4 − Ṽ (vn m)
n (2ζ n t) ∣∣∣∣
n 1/4 > ε ≤ Q n |U w − U ˇw |
µ,ν
n 1/4 > ε ,
t∈[0,1]
sup
w∈T \{∅}
where we have written P ′ for the probability measure under which the sequence (T n ,U n ) n≥1 and
the Brownian snake (,Ö0 ) are defined. From (4.3.12) we get
(
∣ )
P ′ ̂V (vn m
n ) (2ζ n t)
sup
∣ n 1/4 − Ṽ (vn m
n ) (2ζ n t) ∣∣∣∣
n 1/4 > ε −→ 0,
n→∞
t∈[0,1]
and the desired result follows.
The following proposition can be proved using Lemma 4.3.13 and Lemma 4.3.15 in the same
way as Proposition 6.1 in [39].
□
108

Proposition 4.3.16. For every b > 0 and ε ∈ (0,1/10), we can find α,δ ∈ (0,ε) such that
for all n sufficiently large,
(
)
(
)
Q n µ,ν inf V (n) (t) ≥ 2α, sup C (n) (t) + V (n) (t) ≤ ε ≥ 1 − b.
t∈[δ/2,1−δ/2] 2
t∈[0,4δ]∪[1−4δ,1]
Consequently, if K > 0, we have also for all n sufficiently large, for every x ∈ [0,K],
(
)
(
)
P n µ,ν,x inf V (n) (t) ≥ α, sup C (n) (t) + V (n) (t) ≤ ε ≥ 1 − γ 3 b,
t∈[δ,1−δ]
t∈[0,3δ]∪[1−3δ,1]
where the constant γ 3 only depends on µ,ν,K.
4.3.4. Proof of Theorem 4.3.3. The proof below is similar to Section 7 in [39]. We
provide details because the fact that we deal with two-type trees creates nontrivial additional
difficulties.
On a suitable probability space (Ω,P) we can define a collection of processes (e,r z ) z≥0 such
that (e,r z ) is a Brownian snake with initial point z for every z ≥ 0. Recall from section 4.2.3
the definition of (e z ,r z ) and the construction of the conditioned Brownian snake (e 0 ,r 0 ).
Recall that C([0,1], R) 2 is equipped with the norm ‖(f,g)‖ = ‖f‖ u ∨ ‖g‖ u . For every f ∈
C([0,1], R) and r > 0, we set
ω f (r) =
sup |f(s) − f(t)|.
s,t∈[0,1],|t−s|≤r
Let x ≥ 0 be fixed throughout this section and let F be a bounded Lipschitz function. We
have to prove that ( (
E n µ,ν,x F C (n) ,V (n))) −→ E ( F(e 0 ,r 0 ) ) .
n→∞
We may and will assume that 0 ≤ F ≤ 1 and that the Lipschitz constant of F is less than 1.
The first lemma we have to prove gives a spatial Markov property for our spatial trees. We
use the notation of section 5 in [39]. Let recall briefly this notation. We fix a > 0. If (T ,U)
is a mobile and v ∈ T , we say that v is an exit vertex from (−∞,a) if U v ≥ a and U v ′ < a
for every ancestor v ′ of v distinct from v. Notice that, since U v = Uˇv for every v ∈ T 1 , an exit
vertex is necessarily of type 0. We denote by v 1 ,... ,v M the exit vertices from (−∞,a) listed in
lexicographical order. For v ∈ T , recall that T [v] = {w ∈ U : vw ∈ T }. For every w ∈ T [v] we
set
U [v]
w = U vw = U w [v] + U v .
At last, we denote by T a the subtree of T consisting of those vertices which are not strict
descendants of v 1 ,...,v M . Note in particular that v 1 ,...,v M ∈ T a . We also write U a for the
restriction of U to T a . The tree (T a ,U a ) corresponds to the tree (T ,U) which has been truncated
at the first exit time from (−∞,a). The following lemma is an easy application of classical
properties of Galton-Watson trees. We leave details of the proof to the reader.
Lemma 4.3.17. Let x ∈ [0,a) and p ∈ {1,... ,n}. Let n 1 ,...,n p be positive integers such that
n 1 + ... + n p ≤ n. Assume that
)
(M = p, #T [v1],1 = n 1 ,..., #T [vp],1 = n p > 0.
P n µ,ν,x
109

Then, under the probability measure Pµ,ν,x(· n | M = p, #T [v1],1 = n 1 ,... , #T [vp],1 = n p ), and
conditionally on (T a ,U a ), the spatial trees
( ) T [v1] ,U [v 1]
,... ,
(T [vp] ,U [vp])
are independent and distributed respectively according to P n µ,ν,U v1
,...,P n µ,ν,U vp
.
The next lemma is analogous to Lemma 7.1 in [39] and can be proved in the same way using
Theorem 4.3.1.
Lemma 4.3.18. Let 0 < c ′ < c ′′ . Then
(
F
∣ ∣∣E n
sup µ,ν,y
c ′ n 1/4 ≤y≤c ′′ n 1/4
where B = (9(Z q − 1)/(4ρ q )) 1/4 .
(
C (n) ,V (n))) − E
( (
F e By/n1/4 ,r By/n1/4))∣ ∣ −→ 0,
n→∞
We can now follow the lines of section 7 in [39]. Let b > 0. We will prove that for n sufficiently
large,
∣ ( ( ∣∣E n
µ,ν,x F C (n) ,V (n))) − E ( F ( e 0 ,r 0))∣ ∣ ≤ 17b.
We can choose ε ∈ (0,b ∧ 1/10) in such a way that
(4.3.16)
∣ E (F (e z ,r z )) − E ( F ( e 0 ,r 0))∣ ∣ ≤ b,
for every z ∈ (0,2ε). By taking ε smaller if necessary, we may also assume that,
(( ) )
E 3ε sup e 0 (t) ∧ 1 ≤ b, E (ω e 0(6ε) ∧ 1) ≤ b,
(4.3.17)
0≤t≤1
((
E 3ε sup r 0 (t)
0≤t≤1
For α,δ > 0, we denote by Γ n = Γ n (α,δ) the event
{
Γ n =
)
inf V (n) (t) ≥ α,
t∈[δ,1−δ]
)
∧ 1 ≤ b, E (ω r 0(6ε) ∧ 1) ≤ b.
sup
t∈[0,3δ]∪[1−3δ,1]
}
(
)
C (n) (t) + V (n) (t) ≤ ε .
From Proposition 4.3.16, we may fix α,δ ∈ (0,ε) such that, for all n sufficiently large,
(4.3.18) P n µ,ν,x(Γ n ) > 1 − b.
We also require that δ satisfies the following bound
(4.3.19) 4δ(m 1 + 1) < 3ε.
Recall the notation ζ = #T − 1 and B = (9(Z q − 1)/(4ρ q )) 1/4 . On the event Γ n , we have for
every t ∈ [2ζδ,2ζ(1 − δ)],
(4.3.20) V (t) ≥ αB −1 n 1/4 .
This incites us to apply Lemma 4.3.17 with a n = αn 1/4 , where α = αB −1 . Once again, we use
the notation of [39]. We write v1 n,... ,vn M n
for the exit vertices from (−∞,αn 1/4 ) of the spatial
tree (T ,U), listed in lexicographical order. Consider the spatial trees
(
T [vn 1 ] ,U [vn 1 ]) ,... ,
(
T [vn Mn ] ,U [vn Mn ]) .
110

The contour functions of these spatial trees can be obtained in the following way. Set
{
k1 n = inf k ≥ 0 : V (k) ≥ αn 1/4} , l1 n = inf {k ≥ k1 n : C(k + 1) < C(k1)} n ,
and by induction on i,
{
ki+1 n = inf k > li n : V (k) ≥ αn 1/4} , li+1 n = inf { k ≥ ki+1 n : C(k + 1) < C(ki n ) } .
Then ki n ≤ li n < ∞ if and only if i ≤ M n . Furthermore, (C(ki n + t) − C(kn i ),0 ≤ t ≤ ln i − kn i ) is
the contour function of T [vn i ] and (V (ki n + t),0 ≤ t ≤ ln i − kn i ) is the spatial contour function of
(T [vn i ] ,U [vn i ] ). Using (4.3.20), we see that on the event Γ n , all integer points of [2ζδ,2ζ(1 − δ)]
must be contained in a single interval [ki n,ln i ], so that for this particular interval we have
li n − ki n ≥ 2ζ(1 − δ) − 2ζδ − 2 ≥ 2ζ(1 − 3δ),
if n is sufficiently large, P n µ,ν,x a.s. Hence if
E n = {∃i ∈ {1,... ,M n } : l n i − kn i > 2ζ(1 − 3δ)} ,
then, for all n sufficiently large, Γ n ⊂ E n so that
(4.3.21) P n µ,ν,x(E n ) > 1 − b.
As in [39], on the event E n , we denote by i n the unique integer i ∈ {1,... ,M n } such that
l n i −kn i > 2ζ(1 −3δ). We also define ζ n = #T [vn in ] −1 and Y n = U v n
in
. Note that ζ n = (l n i −kn i )/2
so that ζ n > ζ(1 − 3δ). Furthermore, we set
p n = #T [vn in ],1 .
We need to prove a lemma providing an estimate of the probability for p n to be close to n. Note
that p n ≤ n = #T 1 , P n µ,ν,x a.s. Recall that m 1 denotes the mean of µ 1 .
Lemma 4.3.19. For every n sufficiently large,
P n µ,ν,x (Γ n ∩ {p n ≥ (1 − 4δ(m 1 + 1))n}) ≥ 1 − 2b.
Proof : In the same way as in the proof of the bound (4.3.1) we can verify that there exists
a constant β 3 > 0 such that for all n sufficiently large,
)
P µ
(|ζ − (m 1 + 1)n| > n 3/4 ,#T 1 = n ≤ e −nβ3 .
So Lemma 4.3.9 and Proposition 4.3.12 imply that for all n sufficiently large,
(
(4.3.22) P n µ,ν,x |ζ − (m 1 + 1)n| > n 3/4) ≤ b.
Now, on the event Γ n , we have
((
n − p n = # T \ T [vn ]) in ∩ T 1) (
≤ # T \ T [vn ]) in = ζ − ζ n ≤ 3δζ,
since we saw that Γ n ⊂ E n and that ζ n > (1 − 3δ)ζ on E n . If n is sufficiently large, we have
3δ(m 1 + 1)n + 3δn 3/4 ≤ 4δ(m 1 + 1)n so we obtain that
( {
Γ n ∩ ζ ≤ (m 1 + 1)n + n 3/4}) ⊂ (Γ n ∩ {p n ≥ (1 − 4δ(m 1 + 1))n}) ,
for all n sufficiently large. The desired result then follows from (4.3.18) and (4.3.22).
□
111

Let us now define on the event E n , for every t ∈ [0,1],
√
˜C (n) ρq (Z q − 1) C ( ki n (t) =
n
+ 2ζ n t ) − C ( k n )
i n
4
p 1/2 ,
n
( )
Ṽ (n) 9(Zq − 1) 1/4
V ( ki n (t) =
n
+ 2ζ n t )
.
4ρ q
Note that ˜C (n) and Ṽ (n) are rescaled versions of the contour and the spatial contour functions
of (T [vn in ] ,U [vn in ] ). On the event En, c we take ˜C (n) (t) = Ṽ (n) (t) = 0 for every t ∈ [0,1]. Straightforward
calculations show that on the event Γ n , for every t ∈ [0,1],
Set
∣
∣C (n) (t) − ˜C (n) (t) ∣ ≤ ε +
∣
∣V (n) (t) − Ṽ (n) (t) ∣ ≤ ε +
(
(
1 − p1/2 n
n 1/2 )
)
1 − p1/4 n
n 1/4
p 1/4
n
˜Γ n = Γ n ∩ {p n ≥ (1 − 4δ(m 1 + 1))n}.
We then get that on the event ˜Γ n , for every t ∈ [0,1],
∣
∣C (n) (t) − ˜C (n) (4.3.23)
(t) ∣ ≤
(4.3.24)
Likewise, we set
∣
∣V (n) (t) − Ṽ (n) (t) ∣ ≤
sup ˜C (n) (s) + ω eC (n)(6δ),
s∈[0,1]
sup Ṽ (n) (s) + ω eV (n)(6δ).
s∈[0,1]
ε + 4δ(m 1 + 1) sup ˜C (n) (s) + ω eC (n)(6δ),
s∈[0,1]
ε + 4δ(m 1 + 1) sup Ṽ (n) (s) + ω eV (n)(6δ).
s∈[0,1]
Ẽ n = E n ∩ {p n ≥ (1 − 4δ(m 1 + 1))n}.
Lemma 4.3.17 implies that, under the probability measure Pµ,ν,x(· n | Ẽ n ) and conditionally on
the σ-field G n defined by
((
G n = σ T αn1/4 ,U αn1/4) ))
, M n ,
(#T [vn i ],1 , 1 ≤ i ≤ M n ,
the spatial tree (T [vn in ] ,U [vn in ] ) is distributed according to P pn
µ,ν,Y n
(recall that Y n = U v n
in
). Note
that Ẽn ∈ G n , and that Y n and p n are G n -measurable. Thus we have
( (½e En
Eµ,ν,x(
n ˜C(n) ,Ṽ (n)) ∣ ))
∣ Gn
(4.3.25)
(
E n µ,ν,x
(½e En
F ˜C(n) ,Ṽ (n))) = E n µ,ν,x
= E n µ,ν,x
F
(
(½e En
E p µ,ν,Y n
F
(C (p) ,V (p))) p=p n
)
.
From Lemma 4.3.18, we get for every p sufficiently large,
∣ ( ( ∣∣E p
sup µ,ν,y F C (p) ,V (p))) ( (
− E F e By/p1/4 ,r By/p1/4))∣ ∣ ≤ b,
α
2 p1/4 ≤y≤ 3α 2 p1/4
which implies using (4.3.16), since 3αB/2 ≤ 2αB = 2α < 2ε, that for every p sufficiently large,
∣ ∣∣E p
(4.3.26) sup µ,ν,y
α
2 p1/4 ≤y≤ 3α 2 p1/4
(
F
(
C (p) ,V (p))) − E ( F ( e 0 ,r 0))∣ ∣ ∣ ≤ 2b.
112

Furthermore Lemma 4.3.14 implies that
)
P n µ,ν,x
({|n −1/4 Y n − α| > η} ∩ E n −→ 0.
→∞
So we get for every n sufficiently large,
(
)
∣ E n µ,ν,x
(½e En
E p µ,ν,Y n
F
(C (p) ,V (p))) − P n µ,ν,x
p=p n
{
})
≤ 2P n µ,ν,x
(Ẽn ∩ |n −1/4 Y n − α| > α/4
⎛
+ E n µ,ν,x
(4.3.27) ≤ 2b + E n µ,ν,x
∣
⎝ ∣∣E p
sup
α
2 p1/4 ≤y≤ 3α 2 p1/4
⎛
⎝½e En
⎛
⎝½eE n
⎛
µ,ν,y
∣
⎝ ∣∣E p
sup
α
2 p1/4 ≤y≤ 3α 2 p1/4
(Ẽn
)
E ( F ( e 0 ,r 0))∣ ∣ ∣∣
⎞
( (
F C (p) ,V (p))) − E ( F ( e 0 ,r 0))∣ ∣ ⎠
µ,ν,y
⎞
⎠
p=p n
⎞
( (
F C (p) ,V (p))) − E ( F ( e 0 ,r 0))∣ ∣ ⎠
⎠ .
p=p n
⎞
Thus we use (4.3.25), (4.3.26), (4.3.27) and the fact that p n ≥ 1 − 4δ(m 1 + 1)n on Ẽn, to obtain
that for every n sufficiently large,
(
∣
(4.3.28) ∣E n µ,ν,x
(½eE n
F ˜C(n) ,Ṽ (n))) − Pµ,ν,x(Ẽn)E n ( F ( e 0 ,r 0))∣ ∣ ≤ 4b.
From Lemma 4.3.19, we have Pµ,ν,x(Ẽn) n ≥ 1 −2b. Furthermore, 0 ≤ F ≤ 1 so that (4.3.28) gives
(
∣
(4.3.29)
∣E n µ,ν,x F( ˜C
)
(n) ,Ṽ (n) ) − E(F(e 0 ,r 0 )) ∣ ≤ 8b.
On the other hand, since ˜Γ n ⊂ Ẽ n and F is a Lipschitz function whose Lipschitz constant is less
than 1, we have using (4.3.23) and (4.3.24), for n sufficiently large,
∣ (
E n ∣∣F
µ,ν,x
(½e ˜C(n) Γn
,Ṽ (n)) − F
(C (n) ,V (n))∣ )
∣ ((
)
)
(4.3.30)
≤
2ε + E n µ,ν,x
+ E n µ,ν,x
((
4δ(m 1 + 1) sup
s∈[0,1]
˜C (n) (s)
4δ(m 1 + 1) sup Ṽ (n) (s)
s∈[0,1]
)
∧ 1 + ω eC (n)(6δ) ∧ 1
∧ 1 + ω eV (n)(6δ) ∧ 1
By the same arguments we used to derive (4.3.29), we can bound the right-hand side of (4.3.30),
for n sufficiently large, by
((
)
)
b + 2ε + E
+ E
4δ(m 1 + 1) sup
((
s∈[0,1]
e 0 (s)
4δ(m 1 + 1) sup r 0 (s)
s∈[0,1]
∧ 1 + ω e 0(6δ) ∧ 1
)
∧ 1 + ω r 0(6δ) ∧ 1
From (4.3.17) together with (4.3.19), the latter quantity is bounded above by 7b. Since
) (˜Γn ≥ 1 − 2b,
P n µ,ν,x
we get for all n sufficiently large,
(∣ (
E n ∣∣F ˜C(n) µ,ν,x ,Ṽ (n)) − F
(C (n) ,V (n))∣ )
∣ ≤ 9b,
)
.
)
.
113

which implies together with (4.3.29) that for all n sufficiently large,
( (
∣
F C (n) ,V (n))) − E(F ( e 0 ,r 0 ) )∣ ∣ ≤ 17b.
∣E n µ,ν,x
This completes the proof of Theorem 4.3.3
4.3.5. Proof of Theorem 4.2.5. In this section we derive Theorem 4.2.5 from Theorem
4.3.3. We first need to prove a lemma. Recall that if T ∈ T, we set ζ = #T − 1 and we
denote by v(0) = ∅ ≺ v(1) ≺ ... ≺ v(ζ) the list of vertices of T in lexicographical order. For
n ∈ {0,1,... ,ζ}, we set as in [43],
J T (n) = # ( T 0 ∩ {v(0),v(1),... ,v(n)} ) .
We extend J T to the real interval [0,ζ] by setting J T (t) = J T (⌊t⌋) for every t ∈ [0,ζ], and we
set for every t ∈ [0,1]
J T (t) = J T (ζt)
#T 0 .
We also define for k ∈ {0,1,... ,2ζ},
{
}
K T (k) = 1 + # l ∈ {1,... ,k} : C(l) = max C and C(l) is even .
[l−1,l]
Note that K T (k) is the number of vertices of type 0 in the search-depth sequence up to time k.
As previously, we extend K T to the real interval [0,2ζ] by setting K T (t) = K T (⌊t⌋) for every
t ∈ [0,2ζ], and we set for every t ∈ [0,1]
(4.3.31) P n µ,ν,1
K T (t) = K T (2ζt)
#T 0 .
Lemma 4.3.20. The law under P n µ,ν,1 of ( J T (t),0 ≤ t ≤ 1 ) converges as n → ∞ to the Dirac
mass at the identity mapping of [0,1]. In other words, for every η > 0,
(
∣ J T (t) − t ∣ )
> η
(4.3.32) P n µ,ν,1
sup
t∈[0,1]
sup
t∈[0,1]
−→ 0.
n→∞
Consequently, the law under P n µ,ν,1 of ( K T (t),0 ≤ t ≤ 1 ) converges as n → ∞ to the Dirac mass
at the identity mapping of [0,1]. In other words, for every η > 0,
(
∣ KT (t) − t ∣ )
> η
−→ 0.
n→∞
Proof : For T ∈ T, we let v 0 (0) = ∅ ≺ v 0 (1) ≺ ... ≺ v 0 (#T 0 − 1) be the list of vertices of
T of type 0 in lexicographical order. We define as in [43]
G T (k) = # { u ∈ T : u ≺ v 0 (k) } , 0 ≤ k ≤ #T 0 − 1,
and we set G T (#T 0 ) = ζ. Note that v 0 (k) does not belong to the set {u ∈ T : u ≺ v 0 (k)}.
Recall that m 0 denotes the mean of the offspring distribution µ 0 . From the second assertion of
Lemma 18 in [43] there exists a constant ε > 0 such that for n sufficiently large,
(
)
Pµ
n sup |G T (k) − (1 + m 0 )k| ≥ n 3/4 ≤ e −nε .
0≤k≤#T 0
114

Then Lemma 4.3.9 and Proposition 4.3.12 imply that there exists a constant ε ′ > 0 such that
for n sufficiently large,
(
)
(4.3.33) P n µ,ν,1 sup |G T (k) − (1 + m 0 )k| ≥ n 3/4 ≤ e −nε′ .
0≤k≤#T 0
From our definitions, we have for every 0 ≤ k ≤ #T 0 − 1 and 0 ≤ n ≤ ζ,
{G T (k) > n} = {J T (n) ≤ k} .
It then follows from (4.3.33) that, for every η > 0
(
P n µ,ν,1
n −1
sup |J T (((1 + m 0 )k) ∧ ζ) − k| > η
0≤k≤#T 0
Also from the bound (4.3.1) of Lemma 4.3.10 we get for every η > 0,
(∣ ∣∣∣
P n µ,ν,1 n −1 #T 0 − 1 ∣ ) ∣∣∣
> η −→
m 0.
0 n→∞
)
−→ 0.
n→∞
The first assertion of Lemma 4.3.20 follows from the last two convergences.
Let us set j n = 2n − |v(n)| for n ∈ {0,... ,ζ}. It is well known and easy to check by induction
that j n is the first time at which v(n) appears in the search-depth sequence. It is also convenient
to set j ζ+1 = 2ζ. Then we have K T (k) = J T (n) for every k ∈ {j n ,... ,j n+1 − 1} and every
n ∈ {0,... ,ζ}. Let us define a random function ϕ : [0,2ζ] −→ Z + by setting ϕ(t) = n if
t ∈ [j n ,j n+1 ) and 0 ≤ n ≤ ζ, and ϕ(2ζ) = ζ. From our definitions, we have for every t ∈ [0,2ζ],
(4.3.34) K T (t) = J T (ϕ(t)).
Furthermore, we easily check from the equality j n = 2n − |v(n)| that
(4.3.35) sup
∣ ϕ(t) − t 2∣ ≤ max C.
[0,2ζ]
t∈[0,2ζ]
We set ϕ ζ (t) = ζ −1 ϕ(2ζt) for t ∈ [0,1]. So (4.3.35) gives
which implies that for every η > 0,
(
(4.3.36) P n µ,ν,1
sup
t∈[0,1]
sup |ϕ ζ (t) − t| ≤ 1
t∈[0,1] ζ max C,
[0,2ζ]
sup |ϕ ζ (t) − t| > η
t∈[0,1]
)
−→ 0.
n→∞
On the other hand we get from (4.3.34) that K T (t) = J T (ϕ ζ (t)) for every t ∈ [0,1] and thus
∣ KT (t) − J T (t) ∣ ≤ 2 sup ∣ J T (t) − t ∣ + sup |ϕ ζ (t) − t|.
t∈[0,1]
The desired result then follows from (4.3.31) and (4.3.36).
t∈[0,1]
We are now able to complete the proof of Theorem 4.2.5. The proof of (i) is similar to the
proof of the first part of Theorem 8.2 in [39], and is therefore omitted.
Let us turn to (ii). By Corollary 4.2.3 and properties of the Bouttier-di Francesco-Guitter
bijection, the law of λ (n)
M
under Br q (· | #F M = n) is the law under P n µ,ν,1 of the probability
□
115

measure I n defined by
〈I n ,g〉 =
⎛
1
⎝g(0)
#T 0 + ∑ ( ) ⎞
g n −1/4 U v
⎠.
+ 1
v∈T 0
It is more convenient for our purposes to replace I n by a new probability measure I n ′ defined by
〈I n ′ ,g〉 = 1 ∑ )
#T 0 g
(n −1/4 U v .
v∈T 0
Let g be a bounded continuous function. Clearly, we have for every η > 0,
(4.3.37) P n (∣ ∣〈In
µ,ν,1 ,g〉 − 〈I n,g〉 ′ ∣ ) > η −→ 0.
n→∞
Furthermore, we have from our definitions
(4.3.38) 〈I ′ n,g〉 = 1
#T 0 g (
n −1/4) +
∫ 1
0
( )
g n −1/4 V (2ζt) dK T (t),
where the first term in the right-hand side corresponds to v = ∅ in the definition of I n ′ . Then
from Theorem 4.3.3, (4.3.32) and the Skorokhod representation theorem, we can construct on
a suitable probability space, a sequence (T n ,U n ) n≥1 and a conditioned Brownian snake (0,Ö0 ),
such that each pair (T n ,U n ) is distributed according to P n µ,ν,1 , and such that if we write (C n,V n )
for the contour functions of (T n ,U n ), ζ n = #T n − 1 and K n = K Tn , we have,
( )
Vn (2ζ n t)
,K n (t)
n 1/4
−→
n→∞
( ( 4ρq
9(Z q − 1)
) 1/4Ö0 (t), t)
uniformly in t ∈ [0,1], a.s. Now g is Lipschitz, which implies that a.s.,
∫ 1 ( ) ∫ (
1
( ) 1/4Ö0 (4.3.39)
g n −1/4 4ρq
V
∣
n (2ζ n t) dK n (t) − g
(t))
dK n (t)
9(Z q − 1)
∣ −→
0
0
,
n→∞ 0.
Furthermore, the sequence of measures dK n converges weakly to the uniform measure dt on
[0,1] a.s., so that a.s.
∫ (
1
( ) 1/4Ö0 ∫ (
1
( ) 4ρq
4ρq
(4.3.40) g
(t))
dK n (t) −→ g
(t)) 1/4Ö0 dt.
9(Z q − 1)
n→∞ 9(Z q − 1)
0
Then (4.3.39) and (4.3.40) imply that a.s.,
∫ 1 ( )
g n −1/4 V n (2ζ n t) dK n (t) −→
0
n→∞
∫ 1
which together with (4.3.37) and (4.3.38) yields the desired result.
0
g
0
( ( ) 4ρq
(t)) 1/4Ö0 dt,
9(Z q − 1)
Finally, the proof of (iii) from (ii) is similar to the proof of the third part of Theorem 8.2 in
[39]. This completes the proof of Theorem 4.2.5.
116

4.4. Separating vertices in a 2κ-angulation
In this section, we use the estimates of Proposition 4.3.12 to derive a result concerning
separating vertices in rooted 2κ-angulations. Recall that in a 2κ-angulation, all faces have a
degree equal to 2κ.
Let M be a planar map and let σ 0 ∈ V M . Let σ be a vertex of M different from σ 0 . We
denote by S σ 0,σ
M the set of all vertices a of M such that any path from σ to a goes through σ 0.
The vertex σ 0 is called a separating vertex of M if there exists a vertex σ of M different from
σ 0 such that S σ 0,σ
M ≠ {σ 0}. We denote by D M the set of all separating vertices of M.
Recall that U n κ stands for the uniform probability measure on the set of all rooted 2κangulations
with n faces. Our goal is to prove the following theorem.
Theorem 4.4.1. For every ε > 0,
lim
n→∞ U n κ
(
∃ σ 0 ∈ D M : ∃ σ ∈ V M \ {σ 0 }, n 1/2−ε ≤ #S σ 0,σ
M ≤ 2n1/2−ε) = 1.
Theorem 4.4.1 is a consequence of the following theorem. Recall that U n κ denotes the uniform
probability measure on the set of all rooted pointed 2κ-angulations with n faces. If M is a rooted
pointed bipartite planar map, we denote by τ its distinguished point.
Theorem 4.4.2. For every ε > 0,
lim
n→∞ Un κ
(
∃ σ 0 ∈ D M : σ 0 ≠ τ, n 1/2−ε ≤ #S σ 0,τ
M ≤ 2n1/2−ε) = 1.
Theorem 4.4.1 can be deduced from Theorem 4.4.2 but not as directly as one could think.
Indeed the canonical surjection from the set of rooted pointed 2κ-angulations with n faces onto
the set of rooted 2κ-angulations with n faces does not map the uniform measure U n κ to the
uniform measure U n κ . Nevertheless a simple argument allows us to circumvent this difficulty.
Let ˜M r,p be the set of all triples (M,⃗e,τ) where (M,⃗e ) ∈ M r and τ is a distinguished vertex
of the map M. We denote by s the canonical surjection from the set ˜M r,p onto the set M r,p
which is obtained by “forgetting” the orientation of ⃗e. We observe that for every (M,e,τ) ∈ M r,p
# ( s −1 ((M,e,τ)) ) = 2.
Denote by Ũ κ n the uniform measure on the set of all triples (M,⃗e,τ) ∈ ˜M r,p such that M is a
2κ-angulation with n faces. Then the image measure of the measure Ũ κ n under the mapping s is
the measure U n κ. Thus we obtain from Theorem 4.4.2 that
(4.4.1) lim
n→∞ Ũ n κ
(
∃ σ 0 ∈ D M : ∃ σ ∈ V M \ {σ 0 }, n 1/2−ε ≤ #S σ 0,σ
M ≤ 2n1/2−ε) = 1.
On the other hand let p be the canonical projection from the set ˜M r,p onto the set M r . If M
is a 2κ-angulation with n faces, we have thanks to Euler formula
#V M = (κ − 1)n + 2.
Thus the image measure of the measure Ũ n κ under the mapping p is the measure U n κ . This
remark together with (4.4.1) implies Theorem 4.4.1.
The remainder of this section is devoted to the proof of Theorem 4.4.2. We first need to state
a lemma. Recall the definition of the spatial tree (T [v] ,U [v] ) for (T ,U) ∈ Ω and v ∈ T .
117

Lemma 4.4.3. Let (T ,U) ∈ T mob
1 and let M = Ψ r,p ((T ,U)). Suppose that we can find v ∈ T 0
such that T [v],0 ≠ {∅} and U [v] > 0. Then there exists σ 0 ∈ D M such that σ 0 ≠ τ and
#S σ 0,τ
M = #T [v],0 .
Proof : Let (T ,U) ∈ T mob
1 . Write w 0 ,w 1 ,...,w ζ for the search-depth sequence of T 0 (see
Section 4.2.4). Recall from Section 4.2.4 the definition of (U + v ,v ∈ T ) and the construction of
the planar map Ψ r,p ((T ,U)). For every i ∈ {0,1,... ,ζ}, we set s i = ∂ if U + w i
= 1, whereas if
U + w i
≥ 2, we denote by s i the first vertex in the sequence w i+1 ,... ,w ζ−1 ,w 0 ,w 1 ,... ,w i−1 whose
label is U + w i
− 1.
Suppose that there exists v ∈ T 0 such that T [v],0 ≠ {∅} and U [v] > 0. We set
k = min{i ∈ {0,1,... ,ζ} : w i = v},
l = max{i ∈ {0,1,... ,ζ} : w i = v}.
The vertices w k ,w k+1 ,... ,w l are exactly the descendants of v in T 0 . The condition U [v] > 0
ensures that for every i ∈ {k + 1,... ,l − 1}, we have
U + w i
> U + w l
= U + w k
.
This implies that s i is a descendant of v for every i ∈ {k + 1,... ,l − 1}. Furthermore s k = s l
and s i is not a strict descendant of v if i ∈ {0,1,... ,ζ} \ {k,k + 1... ,l}. From the construction
of edges in the map Ψ r,p ((T ,U)) we see that any path from ∂ to a vertex that is a descendant
of v must go through v. It follows that v is a separating vertex of the map M = Ψ r,p ((T ,U))
and that the set T [v],0 is in one-to-one correspondence with the set S v,τ
□
Thanks to Corollary 4.2.4 and Lemma 4.4.3, it suffices to prove the following proposition in
order to get Theorem 4.4.2. Recall the definition of µ κ = (µ κ 0 ,µκ 1 ).
Proposition 4.4.4. For every ε > 0,
lim
n→∞ Pn µ κ ,ν,1
M .
(
∃ v 0 ∈ T 0 : n 1/2−ε ≤ #T [v 0],0 ≤ 2n 1/2−ε , U [v 0] > 0
Proof : For n ≥ 1 and ε > 0, we denote by Λ n,ε the event
{
}
Λ n,ε = ∃ v 0 ∈ T 0 : n 1/2−ε ≤ #T [v0],1 ≤ 2n 1/2−ε , U [v0] > 0 .
Let ε > 0 and α > 0. We will prove that for all n sufficiently large,
(4.4.2) P n µ κ ,ν,1 (Λ n,ε) ≥ 1 − 3α.
We first state a lemma. For T ∈ T and k ≥ 0, we set
Z 0 (k) = #{v ∈ T : |v| = 2k}.
)
= 1.
Lemma 4.4.5. There exist constants β > 0, γ > 0 and M > 0 such that for all n sufficiently
large, (
Pµ n κ inf
γ √ n≤k≤2γ √ Z 0 (k) > β √ n, sup Z 0 (k) < M √ )
n ≥ 1 − α.
n
k≥0
We postpone the proof of Lemma 4.4.5 and complete that of Proposition 4.4.4. To this end,
we introduce some notation. For n ≥ 1, we define an integer K n by the condition
⌈γ √ n ⌉ + K n ⌈n 1/4 ⌉ ≤ ⌊2γ √ n⌋ < ⌈γ √ n ⌉ + (K n + 1)⌈n 1/4 ⌉,
118

and we set for j ∈ {0,... ,K n },
k (n)
j
= ⌈γ √ n ⌉ + j⌈n 1/4 ⌉.
If T ∈ T, we write H(T ) for the height of T , that is the maximal generation of an individual in
T . For k ≥ 0, and N,P ≥ 1 we set
{
}
Z 0 (k,N,P) = # v ∈ T : |v| = 2k,N ≤ #T [v],0 ≤ 2N, H(T [v] ) ≤ 2P .
We denote by Γ n , C n,ε and E n,ε the events
⋂ {
Γ n = β √ ( )
n < Z 0 k (n)
j
0≤j≤K n
C n,ε =
E n,ε =
< M √ }
n ,
⋂ { ( Z 0 k (n)
j
,n 1/2−ε ,n 1/4) > n 1/4} ,
0≤j≤K n
⋂ {
0≤j≤K n
n 1/4 < Z 0 ( k (n)
The first step is to prove that for all n sufficiently large,
j
,n 1/2−ε ,n 1/4) < M √ }
n .
(4.4.3) P n
µ κ (E n,ε) ≥ 1 − 2α.
Since Γ n ∩ C n,ε ⊂ E n,ε , it suffices to prove that for all n sufficiently large,
(4.4.4) Pµ n κ (Γ n ∩ C n,ε ) ≥ 1 − 2α.
We first observe that
∑K n
(4.4.5) P µ κ(Γ n ∩ Cn,ε) c ≤
j=0
P µ κ
(
β √ n < Z 0 ( k (n)
j
Let j ∈ {0,... ,K n }. We have
(
P µ κ β √ ( )
n < Z 0 k (n)
j
< M √ ( n, Z 0 k (n)
j
,n 1/2−ε ,n 1/4) ≤ n 1/4)
(4.4.6)
=
⌊M √ n ⌋
∑
q=⌈β √ n ⌉
P µ κ
( (
Z
k (n)
j
)
< M √ n, Z 0 ( k (n)
j
,n 1/2−ε ,n 1/4) ≤ n 1/4) .
,n 1/2−ε ,n 1/4) ∣ ( ) ) (
≤ n 1/4 ∣∣ Z
0
k (n)
j
= q P µ κ
Z 0 ( k (n)
j
)
)
= q .
Now, under the probability measure P µ κ(· | Z 0 (k (n)
j
) = q), the q subtrees of T above level 2k (n)
j
are independent and distributed according to P µ κ. Consider on a probability space (Ω ′ ,P ′ ) a
sequence of Bernoulli variables (B i ,i ≥ 1) with parameter p n defined by
(
p n = P µ κ n 1/2−ε ≤ #T 0 ≤ 2n 1/2−ε , H(T ) ≤ 2n 1/4) .
Then (4.4.6) gives
(
P µ κ β √ ( )
n < Z 0 k (n)
j
< M √ (
n, Z k (n)
j
,n 1/2−ε ,n 1/4) ≤ n 1/4)
⌊M √ n ⌋
)
∑
≤
B i ≤ n 1/4
(4.4.7)
≤
( q∑
q=⌈β √ n ⌉P ′ i=1
(
⌊M √ n ⌋
∑
q=⌈β √ n ⌉
eexp
− qp nn −1/4
2
)
,
119

where the last bound follows from a simple exponential inequality. However, from Lemma 14 in
[43], there exists η > 0 such that for all n sufficiently large,
p n = P µ κ
(n 1/2−ε ≤ #T 0 ≤ 2n 1/2−ε) − P µ
(n 1/2−ε ≤ #T 0 ≤ 2n 1/2−ε , H(T ) > 2n 1/4)
(
(4.4.8) ≥ P µ κ n 1/2−ε ≤ #T 0 ≤ 2n 1/2−ε) − e −nη .
Under the probability measure P µ κ, we have #T 0 = (κ −1)#T 1 +1 a.s., so we get from Lemma
4.3.9 that there exists a constant c κ such that,
(
(4.4.9) n 1/4−ε/2 P µ κ n 1/2−ε ≤ #T 0 ≤ 2n 1/2−ε) −→ c κ .
n→∞
It then follows from (4.4.8) that
n 1/4−ε/2 p n −→ c κ .
n→∞
From (4.4.7), we obtain that there exists a constant c ′ > 0 such that
(
P µ κ β √ n < Z 0 < M √ (
)
n, Z k (n)
k (n)
j
,n 1/2−ε ,n 1/4) ≤ n 1/4 ≤ M √ ne −c′ n ε ,
j
which together with (4.4.5) implies that
(
P µ κ Γn ∩ Cn,ε
c ) √ ≤ MKn ne
−n ε .
Since K n ∼ γn 1/4 as n → ∞, we get from Lemma 4.3.9 that, for all n sufficiently large,
(4.4.10) Pµ n ( κ Γn ∩ Cn,ε
c ) P µ κ(Γ n ∩ Cn,ε)
c
≤
P µ κ(#T 1 = n) ≤ α.
On the other hand, Lemma 4.4.5 implies that
The bound (4.4.4) now follows.
Set
I n =
For (p 0 ,... ,p Kn ) ∈ I n , we set
E n,ε (p 0 ,... ,p Kn ) =
P n
µ κ(Γ n) ≥ 1 − α.
{
⌈n 1/4 ⌉, ⌈n 1/4 ⌉ + 1,... , ⌊M √ n⌋} Kn+1
.
⋂ { ( Z 0 k (n)
j
,n 1/2−ε ,n 1/4) }
= p j .
0≤j≤K n
On the event E n,ε (p 0 ,... ,p Kn ) for every j ∈ {1,... ,K n }, let v j 1 ≺ ... ≺ vj p j
be the list in
lexicographical order of those vertices v ∈ T at generation 2k (n)
j
such that n 1/2−ε ≤ #T [v],0 ≤
2n 1/2−ε and H(T [v] ) ≤ 2n 1/4 . Note that for every j,j ′ ∈ {1,... ,K n } such that j < j ′ , we have
for every i ∈ {1,... ,p j },
{
max |v j i v| : v ∈ T [vj]} i ≤ 2k (n)
j
+ 2n 1/4 < 2k (n)
j
. ′
Then, it is not difficult to check that under P µ κ ,ν,1(· | E n,ε (p 0 ,... ,p Kn )), the spatial trees
{(T [vj i ] ,U [vj i ] ), i = 1,... ,p j , j = 1,... ,K n } are independent and distributed according to the
probability measure P µ κ ,ν,0(· | n 1/2−ε ≤ #T 0 ≤ n 1/2−ε , H(T ) ≤ 2n 1/4 ). Set
π n = P µ,ν,0
(U > 0 | n 1/2−ε ≤ #T 0 ≤ 2n 1/2−ε , H(T ) ≤ 2n 1/4) .
120

Since the events E n,ε (p 0 ,... ,p Kn ), (p 0 ,... ,p Kn ) ∈ I n are disjoints we have,
∑
P µ κ ,ν,1(E n,ε ∩ Λ c n,ε) = P µ κ(E n,ε (p 0 ,... ,p Kn ))
(p 0 ,...,p Kn )∈I n
}
)
× P µ κ ,ν,1
({U [vj i ] ≤ 0 : 0 ≤ j ≤ K n ;1 ≤ i ≤ p j | E n,ε (p 0 ,...,p Kn )
∑
=
(4.4.11)
≤
(p 0 ,...,p Kn )∈I n
P µ κ(E n,ε (p 0 ,... ,p Kn ))(1 − π n ) p 0+...+p Kn
P µ κ(E n,ε )(1 − π n ) ⌈n1/4 ⌉(K n+1)
≤ (1 − π n ) ⌈n1/4 ⌉(K n+1) .
Now, from Lemma 14 in [43] and (4.4.9) there exists c ′′ > 0 such that for all n sufficiently large,
π n ≥ P µ,ν,0
(U > 0 | n 1/2−ε ≤ #T 0 ≤ 2n 1/2−ε) − c ′′ n 1/2 e −nη ,
where η was introduced before (4.4.8). Since under the probability measure P µ κ, we have #T 0 =
1 + (κ − 1)#T 1 a.s., we get from Proposition 4.3.12 that there exists a constant c > 0 such that
for all n sufficiently large,
π n ≥
c
n 1/2−ε.
So, there exists a constant c ′ > 0 such that (4.4.11) becomes for all n sufficiently large,
(4.4.12) P µ κ ,ν,1
(
En,ε ∩ Λ c n,ε)
≤ e
−c ′ n ε .
Finally (4.4.12) together with Lemma 4.3.9 implies that if n is sufficiently large,
Using also (4.4.3), we obtain our claim (4.4.2).
P n µ κ ,ν,1
(
En,ε ∩ Λ c n,ε)
≤ α.
Proof of Lemma 4.4.5 : Under P µ κ, Z 0 = (Z 0 (k),k ≥ 0) is a critical Galton-Watson
process with offspring law µ κ 0,κ supported on (κ − 1)Z + and defined by
µ κ 0,κ ((κ − 1)k) = µκ 0 (k), k ≥ 0.
Note that the total progeny of Z 0 is #T 0 and recall that under the probability measure P µ κ,
we have #T 0 = 1 + (κ − 1)#T 1 a.s.
Define a function L = (L(t),t ≥ 0) by interpolating linearly Z 0 between successive integers.
For n ≥ 1 and t ≥ 0, we set
l n (t) = √ 1 L(t √ n). n
Denote by l(t) the total local time of e at level t that is,
1
l(t) = lim
ε→0 ε
∫ 1
0½{t≤e(s)≤t+ε} ds,
where the convergence holds a.s. From Theorem 1.1 in [19], the law of (l n (t),t ≥ 0) under Pµ n κ
converges to the law of (l(t),t ≥ 0). So we have
lim inf P n
n→∞
µ κ (
)
inf l n(t) > β, sup l n (t) < M
γ≤t≤2γ
t≥0
(
≥ P
inf l(t) > β, sup l(t) < M
γ≤t≤2γ t≥0
)
.
□
121

However we can find β > 0, γ > 0 and M > 0 such that
(
)
P inf l(t) > β, sup l(t) < M ≥ 1 − α
γ≤t≤2γ t≥0
2 ,
and the desired result follows.
□
122

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