Mikhail SODIN

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On the number of components of random nodal sets
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joint work with Fedor Nazarov (Kent)
Puerto de la Cruz, Tenerife
March, 2012
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The first half of Hilbert’s 16th problem: bounds and possible configurations of
connected components of the zero set of an algebraic polynomial in R m .
We study its statistical version and find the order of growth of the typical
number of components of zero sets of smooth random functions of several real
variables.
The primary examples are
• various ensembles of Gaussian real-valued polynomials (algebraic or
trigonometric) of large degree (“Riemannian case”)
• smooth Gaussian functions on the Euclidean space with translation-invariant
distribution. (“Euclidean case”)
Major difficulty/interest: “non-locality” of the problem (no integral formulas,
like Jensen or Kac-Rice).
Remark: upper bounds can be obtained from counting critical points. Lower
bounds are more involved.
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Nodal portrait of a gaussian spherical harmonic of degree 40
(figure by Alex Barnett)
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Part I: Euclidean case
F : R m → R a random Gaussian function with translation-invariant
distribution:
for each k ∈ N, each u1, ..., uk ∈ R m , and each v ∈ R m , the random
vectors F (u1), ..., F (uk) and F (u1 + v), ..., F (uk + v) have the same
normal distribution.
Then E {F (u)F (v)} = k(u − v),
k(u) = e 2πiu·λ
dρ(λ) =
✫
R m
R m
ρ ∈ M + (R m ) is the spectral measure of F .
We always assume: for some p > 2,
R m
=⇒ a.s. the random function F is C p -smooth.
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cos (2πu · λ) dρ(λ) .
|λ| 2p dρ(λ) < ∞.
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Notation: N(R; F ) the number of connected components of the zero set
Z(F ) = F −1 {0} that are contained in the open ball B(R) = {x: |x| < R}
Theorem I: Suppose that the spectral measure ρ has no atoms and is not
supported by a hyperplane. Then there exists a constant ν(ρ) ≥ 0 such
that
✫
lim
R→∞
N(R; F )
volB(R)
= ν(ρ) a.s. , and lim
R→∞
EN(R; F )
volB(R)
= ν(ρ) .
Furthermore, the limiting constant ν(ρ) is positive provided that
(∗) ∃ a compactly supported symmetric measure µ with spt(µ) ⊂ spt(ρ)
s.t. the zero set of its Fourier transform µ has a bounded component.
The proof does not give much information about the value of the constant ν(ρ);
a huge discrepancy between lower bounds extracted from that proof and upper
bounds obtained by computing the mean number of critical points :(
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How to check condition (∗)?
A simple and crude sufficient condition which holds in many examples:
• the closed support of ρ contains a sphere centered at the origin.
In this case, one can take µ = the Lebesgue measure on the sphere. Then
µ is radially symmetric and vanishes on concentric spheres with radii
tending to infinity.
Using a little functional analysis, one can go further:
• the closed support of ρ contains an open subset of a sphere centered at
the origin
In the planar case (m = 2), one can show that condition (∗) holds
whenever
• the support of ρ contains a compact subset that cannot be covered by
finitely many straight lines.
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Part II: Riemannian case
X a smooth compact m-dimensional Riemannian manifold without
boundary. Normalization of the volume form: vol(X) = 1.
HL a family of real finite-dimensional Hilbert spaces of smooth functions
on X, dimHL → ∞ as L → ∞.
KL(x, y) the reproducing kernel of the space HL:
✫
f(y) = 〈f( . ), KL( . , y)〉HL , f ∈ HL, y ∈ X .
We always assume: the function x ↦→ KL(x, x) does not vanish on X.
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Gaussian functions on X:
The space HL generates a random Gaussian function
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fL(x) = ξkek(x), x ∈ X ,
where
ek is an orthonormal basis in HL, and ξk are independent
standard Gaussian random variables.
The covariance of the Gaussian function fL:
E fL(x)fL(y) = ek(x)ek(y) = KL(x, y)
The distribution of fL does not depend on the choice of the orthonormal
basis
ek in HL.
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Normalization:
The functions fL are normalized if Ef 2 L (x) = KL(x, x) = 1, x ∈ X.
Wlog, we assume that the functions fL are normalized.
Otherwise, replace the functions fL and the kernel KL by
✫
fL(x) = fL(x)
Ef 2 L (x) , KL(x, y) =
KL(x, y)
KL(x, x) · KL(y, y) .
This normalization changes the Hilbert spaces HL but the zero sets of
the Gaussian functions fL and fL remain the same.
In basic examples, the function x ↦→ KL(x, x) is constant (that is, the
norm of the point evaluation in HL does not depend on the point), so the
normalization boils down to computation of that constant.
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Scaling (blowing up with the coefficient L ≫ 1):
TxX the tangent space at x;
exp x : V → X the exponential map,
V ⊂ TxX neighbourhood of the origin;
Ix : Rm → Tx(X) a linear Euclidean isometry;
Φx = expx ◦Ix : U → X, Φx(0) = x,
where U ⊂ Rm neighbourhood of the origin.
To scale the covariance kernel KL around the point x ∈ X in L times,
we let Φx,L(u) def
= Φx(L−1u), and put
✫
Kx,L(u, v) def
= KL (Φx,L(u), Φx,L(v)) , u, v ∈ R m .
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Translation-invariant local limits:
Definition: The Gaussian ensemble (fL) has translation-invariant local
limits as L → ∞ if for a.e. x ∈ X, there exists a Hermitean positive
definite function kx : R m → R 1 , such that for each R < ∞,
✫
lim
L→∞
sup
|u|,|v|≤R
|Kx,L(u, v) − kx(u − v)| = 0 .
The limiting kernels kx(u − v) are covariance kernels of
translation-invariant Gaussian functions Fx : R m → R 1 (x ∈ X).
They are the Fourier transforms of probability measures ρx on R m ,
symmetric w.r.t. the origin.
We call the function Fx the local limiting function, and the measure ρx
the local limiting spectral measure of the family fL at the point x.
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Technical assumptions:
C2-smoothness: the Gaussian ensemble (fL) is C2-smooth if
lim
L→∞ EfL 2 C(X) + L−1∇fL 2 C(X) + L−2∇ 2 fL 2
C(X) < ∞ .
Non-degeneracy: there is a set X ′ ⊂ X of full volume (that is,
vol(X \ X ′ ) = 0) such that
inf
x∈X ′
inf 〈v, λ〉
v=1
2 dρx(λ) > 0 .
✫
R m
In the case when the limiting spectral measure ρx does not depend on
x ∈ X this says that the measure ρ is not supported by a hyperplane.
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Notation: N(fL) the number of components of the zero set Z(fL)
Theorem II: Suppose that (fL) is a C 2 -smooth Gaussian ensemble on
X, which has translation-invariant local limits a.e. on X. Suppose that
the local limiting spectral measures ρx have no atoms and satisfy the
non-degeneracy condition from the previous slide.
Then the function x ↦→ ν(ρx) belongs to L∞ (X), and
lim
L→∞ E
L −m
N(fL) − ν(ρx)dvol(x) = 0 .
Here, ν(ρx) is a limiting constant from Theorem I (Euclidean case).
Remark: One can see that
X ν(ρx) dvol(x) does not depend on the choice
of the Riemannian metrics, only the smooth structure on X matters.
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X
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Local version of Theorem II:
The limiting constant ν(ρx) can be recovered by a double-scaling limit:
for a.e. x ∈ X and for each ɛ > 0,
lim
R→∞ lim
L→∞ P
1
volB(R) Nx, R
L ; fL − ν(ρx) > ɛ = 0
where N x, R
L ; fL is a number of connected components of the zero set
Z(fL) containing in the open ball centered at x of radius R/L,
volB(R) is the Euclidean volume of the ball of radius R in Rm .
Theorem II is “an integrated version” of the local result.
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Part III. Related works:
T.L.Malevich (1973): non-trivial lower bound for the mean number of
components. She considered C 2 -smooth Gaussian functions F on R 2 with
positive covariance function and proved that 0 < c ≤ EN(R; F )/R 2 ≤ C < ∞.
Her proof uses the positivity property of the covariance function, which in
many instances does not hold.
E.Bogomolny and C.Schmit (2002): bond percolation model for description of
the zero set of translation-invariant Gaussian function F on R 2 whose spectral
measure is a Lebesgue measure on the unit circumference. Their model ignores
slow decaying correlations between values of F at different points, and is far
from being rigorous. Computations based on that model are well supported by
numerics.
It is not clear whether their approach can be extended to other spectral
measures or to higher dim.
Challenge: “hidden universality law” that provides the rigorous foundation for
the work done by Bogomolny and Schmit.
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Related works (continuation):
F.Nazarov and M.S. (2009): for the Gaussian ensemble of spherical harmonics
of large degree on the two-dimensional sphere,
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lim
L→∞ P L −2 N(fL) − υ > ɛ < C(ɛ)e −c(ɛ) dim H L ,
with some υ > 0. The limiting function for this ensemble is the one considered
by Bogomolny and Schmit. The exponential concentration of N(fL)/L 2 is
surprising since the Gaussian ensemble treated there has slow decaying
correlations.
We were unable to prove the exponential concentration for other ensembles
considered here. The difficulty is caused by components of small diameter,
which do not exist when fL is an eigenfunction of the Laplacian. Even in the
univariate case, the question about exponential concentration remains open; cf.
Tsirelson’s lecture notes, Tel Aviv University, Fall 2010
http://www.tau.ac.il/~tsirel/Courses/Gauss3/main.html
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Part IV. Examples
1. Trigonometric ensemble X = T m (m-dim torus)
Hn,m ⊂ L 2 (T m ), subspace of trigonometric polynomials in m variables of
degree ≤ n in each of the variables.
The repro-kernel (= covariance): the Dirichlet kernel
m sin [π(2n + 1)(xj − yj)]
Kn,m(x, y) =
(2n + 1) sin [π(xj − yj)]
✫
j=1
scaling parameter L = n (the degree).
After scaling, covariance converges together with partial derivatives of
m sin 2πuj
any order to the limiting kernel k(u − v), k(u) =
.
2πuj
Hence, the limiting spectral measure is the normalized Lebesgue measure
on the cube [−1, 1] m ⊂ R m , which meets all our assumptions.
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j=1
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2. Spherical ensemble: X = S m (m-dim sphere)
Hn,m ⊂ L 2 (S m ) subspace spanned by polynomials in m + 1 variables of
degree ≤ n, restricted on S m .
Repro-kernel in Hn,m : c(n, m)P
P (α,β)
n
✫
m m ( 2 , 2 −1)
n
(x · y), x, y ∈ S m ;
Jacobi polynomials of degree n and of index (α, β); i.e.,
polynomials orthogonal on [−1, 1] with the weight (1 − x) α (1 + x) β .
m m m (
Mehler-Heine asymptotics: lim n− 2 2 , 2 P
n→∞ −1) z z
n cos =
n 2
(z) is Bessel’s function, the convergence is locally uniform in C.
J m
2
Scaling parameter L = n (the degree)
− m
2 J m
2 (z),
Limiting spectral measure ρ = Lebesgue measure on the unit ball in R m .
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3. Kostlan ensemble: Homogeneous polynomials of degree n in m + 1 on
X = PR m (m-dim projective space).
The scalar product 〈f, g〉 =
✫
f(X) =
|J|=n
|J|=n
fJX J , g(X) =
n
fJgJ, where
J
|J|=n
J = (j0, j1, j2, ... jm), |J| = j0 + j1 + j2 + ... + jm, n
= J
gJx J , X J = x j0
0 x j1
1 x j2
2 ... x jm m ,
n!
j0!j1!j2! ... jm! .
Complexification: after continuation of the homogeneous polynomials f and g
to C m+1 , the scalar product coincides with the one in the Fock-Bargmann
space 〈f, g〉 = cm f(Z)g(Z)e −|Z|2
dvol(Z).
C m+1
This is the only unitary invariant Gaussian ensemble of homogeneous
polynomials.
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Kostlan ensemble (continuation):
In homogeneous coordinates, the covariance kernel equals X·Y
n. chart x0 = y0 = 1, we get 1+(x·y) √
1+|x| 2 √ 1+|y| 2
The features:
✫
|X| |Y |
n. In the
• L = √ n (square root of the degree, not the degree, as in previous examples)
• very rapid decay of the covariance away from the diagonal.
The limiting spectral measure is the Gaussian measure on R m with the density
exp (x · y) − 1
2 |x|2 − 1
2 |y|2 . Once again, Theorem II is applicable.
Asymptotic distribution of of the number of components in Kostlan ensemble
was recently studied by D.Gayet and J-Y.Welschinger, and by P.Sarnak and
I.Wigman.
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Part V: Steps in the proof of Theorem I
Step 1: Some integral geometry
Notation: N(u, r; F ) the number of connected components of Z(F )
contained in the open ball B(u, r)
¯N(u, r; F ) the number of connected components of Z(F ) that intersect
the closed ball B(u, r)
Lemma: For 0 < r < R < ∞,
N(u, r; F )
du ≤ N(R; F ) ≤
volB(r)
✫
B(R−r)
We apply this with 1 ≪ r ≪ R.
B(R+r)
¯N(u, r; F )
volB(r) du
Notation: (τvF )(u)=F (u + v) (translation). Then N(u, r; F )=N(r; τuF )
Observe: ¯ N(r; F ) − N(r; F ) ≤ N(r; F ) def
= # of critical pts of F ∂B(r)
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to be continued on the next slide
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continuation
✫
1 − o(1)
N(r; τuF )
volB(R − r) B(R−r) volB(r)
1 + o(1)
≤
volB(R + r)
B(R+r)
du ≤ N(R; F )
volB(R)
N(r; τuF ) + N(r; τuF )
volB(r)
Note: LHS and RHS are spatial averages of translations
Step 2: Metric transitivity
Fomin-Grenander-Maruyama: the spectral measure ρ has no atoms
du, for R ≫ r
=⇒ translations τu act metric-transitively on the Borel σ-algebra of F (that is,
all invariant sets have probability 0 or 1)
=⇒ the distribution of N(r; F ) is also metric transitive
=⇒ ergodic theorem can be applied
=⇒ for fixed r, a.s.,
lim
R→∞
N(R; F )
volB(R)
EN(r; F )
≥ , lim
volB(r) R→∞
N(R; F )
volB(R)
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≤ EN(r; F )
volB(r)
+ EN(r; F )
volB(r)
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Step 3: The Kac-Rice bound for the number of critical pts:
Lemma: EN(r; F ) volm−1∂B(r).
=⇒ a.s., lim R
✫
N(R; F )
volB(R) exists and equals lim r
EN(r; F )
volB(r)
=: ν(ρ)
Step 4: Positivity of ν(ρ): We need to show: for some r0 > 0, EN(r0; F ) > 0.
Standard Gaussian Lemma: Suppose µ is a compactly supported measure
with spt(µ) ⊂ spt(ρ). Then for each ball B ⊂ R m and each ɛ > 0,
P F − µ C( ¯ B) < ɛ > 0.
By assumption (∗) in Theorem I, ∃ such a measure µ with Z(µ) having a
bounded connected component.
By real analyticity of µ, this component is isolated.
Choosing ɛ small enough, we get P N(r0; F ) > 0 > 0 for some r0
=⇒ EN(r0; F ) > 0.
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The End
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