arbres alÃ©atoires, conditionnement et cartes planaires - DMA - Ens

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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
Page 1: THÈSE DE DOCTORAT DE L’UNIVERSIT
Page 5: Table des matières Chapitre 1. Int
Page 8 and 9: t ∈ [0,2(#A − 1)] entier, on d
Page 10 and 11: Remarquons qu’un arbre planaire p
Page 12 and 13: Théorème 1.2.2. Soit Θ une mesur
Page 14 and 15: Rappelons que si s ∈ [0,1], alors
Page 16 and 17: On définit ensuite W [s] à partir
Page 18 and 19: 1.5. Arbres spatiaux et cartes plan
Page 20 and 21: Si de plus U v ≥ 1 pour tout v
Page 22 and 23: Fig. 2. Construction d’une carte
Page 24 and 25: Théorème 1.6.2. Soit q une suite
Page 27 and 28: CHAPITRE 2 Regenerative real trees
Page 29: andom ordering to the discrete tree
Page 33 and 34: first generation. Then, if F : A k
Page 35 and 36: Let us now set I t = inf {X s : 0
Page 37 and 38: Let us fix k ≥ 1 with µ ε (k) >
Page 39 and 40: The bound (2.3.6) implies ( ∑ ∞
Page 41 and 42: Now, thanks to Proposition 2.3.6,
Page 43 and 44: Now, Corollary 2.3.7 and Propositio
Page 45 and 46: Proposition 2.4.4. For every ε > 0
Page 47 and 48: Since f is bounded and Xu 1/n ≤ L
Page 49: where θ 1 ,... ,θ p denote the co
Page 52 and 53: This definition should become clear
Page 54 and 55: the geometric distribution, this gi
Page 56 and 57: Brownian motion in [0, ∞[ started
Page 58 and 59: for every u ∈ T and l ∈ [0,h u
Page 60 and 61: if r ∈ [0,σ], and Ŵ r [s] = 0 i
Page 62 and 63: where the last equality follows fro
Page 64 and 65: where c(ε,δ) = N q (B ε,δ ) doe
Page 66 and 67: Similarly, N 0 (½{W>−ε}e −σ
Page 68 and 69: To simplify notation, we have writt
Page 70 and 71: N = ∑ i∈I δ ω i with intensit
Page 72 and 73: It remains to study ε −4 N 0 (½
Page 74 and 75: Proof of Theorems 3.1.1 and 3.1.2 :
Page 76 and 77: For every h > 0, we set N h 0 = N 0
Page 78 and 79: Now note that the range of the Brow
Page 80 and 81:
and conditionally given W α , the
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for every i ∈ I and s ≥ 0. For
Page 84 and 85:
and then, for every u ∈ T \{∅},
Page 86 and 87:
Q θ ε As a consequence of Lemma 3
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From (3.5.3) and Lemma 3.5.3, it th
Page 90 and 91:
ipartite planar maps are in one-to-
Page 92 and 93:
Let us turn to the special case of
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Write P for the probability measure
Page 96 and 97:
From [43], we know that µ 1 has sm
Page 98 and 99:
(ii) The law of the random measure
Page 100 and 101:
which means that k is the time of t
Page 102 and 103:
4.3.2. Estimates for the probabilit
Page 104 and 105:
Proof : The proof of this lemma is
Page 106 and 107:
Recall that p ≥ 1 is such that µ
Page 108 and 109:
such that each pair (T n ,U n ) is
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Then, under the probability measure
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Let us now define on the event E n
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which implies together with (4.3.29
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measure I n defined by 〈I n ,g〉
Page 118 and 119:
Lemma 4.4.3. Let (T ,U) ∈ T mob 1
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where the last bound follows from a
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However we can find β > 0, γ > 0
Page 124:
[25] Evans, S.N., Winter, A. Subtre
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arbres alÃ©atoires, conditionnement et cartes planaires - DMA - Ens

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