Russo-Seymour-Welsh estimates for the Kostlanensemble of random polynomials

Dmitry Beliaev

Mathematical InstituteUniversity of Oxford

5 December 2017

Random geometries / Random topologies

Laplace Eigenfunctions: Chladni figures

Interest in the nodal lines of Laplace eigenfunctions has a very longhistory and goes back to Hooke (XVII century) and Chladni (XVIIIcentury) who observed nodal lines on a vibrating plate.

Figure: Chladni figures (MIT Physiscs Demos)

Berry’s conjecture

In 1977 M. Berry conjectured that high energy eigenfunctions inthe chaotic case have statistically the same behaviour as randomplane waves. (Figures from Bogomolny-Schmit paper)

Figure: Nodal domains of an eigenfunctionof a stadium

Figure: Nodal domains of arandom plane wave

Random Ensembles

(M, g) Riemannian manifold, ��n

+ �n

�n

= 0Monochromatic weave: random combination of eigenfunctionswith close eigenvalues

f1,T =X

T�pT<�

i

<T

cn

�n

, ci

i.i.d. normal

Band-limited: random combination of eigenfunctions for a widerange of frequencies

f↵,T =X

↵T<�i

<T

cn

�n

, ci

i.i.d. normal

Spherical ensembles

For sphere eigenfunctions are spherical harmonics. There are2n + 1 spherical harmonics of degree n, eigenvalue n(n + 1).

Figure: Random spherical harmonicof degree 80, ↵ = 1

Figure: Real Fubini-Study ensembleof degree 80, ↵ = 0

Scaling limits of spherical ensembles

It is possible to pass to the (universal) local scaling limit as degree(energy) goes to infinity.

Figure: The random plane wave isthe scaling limit of the randomspherical harmonic

Figure: The band-limited function isthe scaling limit of the realFubini-Study

Scaling Limit: Random Plane Wave

Two ways to (informally) think of the random plane wave

A “random” or “typical” solution of Helmholtz equation�f + k2f = 0

A random superposition of all possible plane waves with thesame frequency k

Formal definition:

A Gaussian field f (z) with covariance kernel

K (z ,w) = E[f (w)f (w)] = J0(k |z � w |)

Random series

f (z) = f (re i✓) = ReX

an

J|n|(r)ein✓

Nodal Lines of Gaussian Spherical Harmonic

Theorem (Berard, 1985)

For Gaussian spherical harmonic gn

of degree n

EL(gn

) = ⇡p2�

n

=p2⇡n + O(1)

Theorem (Nazarov and Sodin, 2007)

Let gn

be Gaussian spherical harmonic of degree n. Then there is apositive constant a such that

P⇢����

N(gn

)

n2� aVol(S2)

���� > ✏

� C (✏)e�c(✏)n

where C (✏) and c(✏) are positive constant depending on ✏ only.

Bogomolny-Schmit Percolation Model

They proposed that the nodal lines form a perturbed square lattice

Picture from Bogomolny-Schmit paper.

Bogomolny-Schmit Percolation Model

Using this analogy we can think of the nodal domains aspercolation clusters on the square lattice. This leads to theconjecture that a = (3

p3� 5)/⇡ ⇡ 0.0624

Picture from Bogomolny-Schmit paper.

Is It Really True?

Numerical results (Nastasescu (2011), Konrad (2012),B.-Kereta (2013)) show that the number of nodal domainsper unit area is 0.0589 instead of 0.0624 predicted byBogomolny-Schmit.

Number of clusters per vertex is a non-universal quantity inpercolation, it is lattice dependent. Global properties shouldbe universal i.e. lattice independent.

Numerical evidence that many global ‘universal’ observables(crossing probabilities, decay rate for the area of nodaldomains, one-arm exponent) match percolation predictions.

O↵-critical models

O↵-critical percolation is a model for excursion and level sets

Figure: Excursion sets for levels 0 (nodal domains) and level 0.1

O↵-critical models

O↵-critical percolation is a model for excursion and level sets

Figure: Excursion sets for levels 0 (nodal domains) and level 0.1

Kostlan ensemble

Figure: Kostlan ensemblen = 300

Kostlan or complex Fubini-Studyensemble of homogeneouspolynomials R3 (or S2)

f (x) =X

|J|=n

aJ

s✓n

J

◆xJ

Covariance kernel cosn(d(x , y))

Natural measure on spaceof homogeneous polynomials

Nodal domains is a natural modelfor real projective curve

Bargmann-Fock Random Function

Theorem (Be↵ara-Gayet 2016)

Russo-Seymour-Welsh estimate for Bargmann-Fock randomfunction.

Figure: Nodal domains ofBargmann-Fock function

Bargmann-Fock Gaussian function isthe scaling limit of Kostlan ensemble

f (x) =X

ai ,j

1pi !j!

x i1xj

2e�|x |2/2

Covariance kernel

K (x , y) = e�|x�y |2/2

Important:positive, symmetric, fast decaying

Russo-Seymour-Welsh for Kostlan Ensemble

Theorem (B.-Muirhead-Wigman)

There is RSW estimate for nodal domains and nodal lines ofKostlan ensemble which is uniform in polynomial degree and‘conformal type’ of a domain.Also true for general symmetric fields with su�ciently fast decay ofcorrelation.

General result

Theorem (Abstract RSW)

Let fn

be a collection of Gaussian fields with covariance kernels n

.We assume that there are s

n

! 0 such that

1 Fields fn

and kernels n

are symmetric

2 Fields are smooth and non-degenerate

3 n

! K1 locally uniformly on scale sn

4 Asymptotically non-negative correlations

5 Uniform rapid decay of correlations

General result

Theorem (Abstract RSW)

Let fn

be a collection of Gaussian fields with covariance kernels n

.We assume that there are s

n

! 0 such that

1 Fields fn

and kernels n

are symmetric

2 Fields are smooth and non-degenerate

3 n

! K1 locally uniformly on scale sn

4 Asymptotically non-negative correlations

5 Uniform rapid decay of correlations

General result

Theorem (Abstract RSW)

Let fn

be a collection of Gaussian fields with covariance kernels n

.We assume that there are s

n

! 0 such that

1 Fields fn

and kernels n

are symmetric

2 Fields are smooth and non-degenerate

Fields fn

2 C 3 and n

2 C 6. Hessian of n

at 0 is positive definite

3 n

! K1 locally uniformly on scale sn

4 Asymptotically non-negative correlations

5 Uniform rapid decay of correlations

General result

Theorem (Abstract RSW)

Let fn

be a collection of Gaussian fields with covariance kernels n

.We assume that there are s

n

! 0 such that

1 Fields fn

and kernels n

are symmetric

2 Fields are smooth and non-degenerate

3 n

! K1 locally uniformly on scale sn

4 Asymptotically non-negative correlations

5 Uniform rapid decay of correlations

General result

Theorem (Abstract RSW)

Let fn

be a collection of Gaussian fields with covariance kernels n

.We assume that there are s

n

! 0 such that

1 Fields fn

and kernels n

are symmetric

2 Fields are smooth and non-degenerate

3 n

! K1 locally uniformly on scale sn

Let � : R2 ! S2 exponential map (at some point), thenKn

(x , y) = n

(�(sn

x),�(sn

y)) ! K1(x , y) locally uniformlytogether with the first four derivatives.

4 Asymptotically non-negative correlations

5 Uniform rapid decay of correlations

General result

Theorem (Abstract RSW)

Let fn

be a collection of Gaussian fields with covariance kernels n

.We assume that there are s

n

! 0 such that

1 Fields fn

and kernels n

are symmetric

2 Fields are smooth and non-degenerate

3 n

! K1 locally uniformly on scale sn

4 Asymptotically non-negative correlations

5 Uniform rapid decay of correlations

General result

Theorem (Abstract RSW)

Let fn

be a collection of Gaussian fields with covariance kernels n

.We assume that there are s

n

! 0 such that

1 Fields fn

and kernels n

are symmetric

2 Fields are smooth and non-degenerate

3 n

! K1 locally uniformly on scale sn

4 Asymptotically non-negative correlations

Negative part of n

is o(s12+✏n

). Exponent 12 is not optimal, couldbe replaced by 8 (assuming convergence of six derivatives). Seealso Be↵ara-Gayet.

5 Uniform rapid decay of correlations

General result

Theorem (Abstract RSW)

Let fn

be a collection of Gaussian fields with covariance kernels n

.We assume that there are s

n

! 0 such that

1 Fields fn

and kernels n

are symmetric

2 Fields are smooth and non-degenerate

3 n

! K1 locally uniformly on scale sn

4 Asymptotically non-negative correlations

5 Uniform rapid decay of correlations

General result

Theorem (Abstract RSW)

Let fn

be a collection of Gaussian fields with covariance kernels n

.We assume that there are s

n

! 0 such that

1 Fields fn

and kernels n

are symmetric

2 Fields are smooth and non-degenerate

3 n

! K1 locally uniformly on scale sn

4 Asymptotically non-negative correlations

5 Uniform rapid decay of correlations

|n

(x , y)| . (d(x , y)/sn

)�18�✏) for d(x , y) & sn

. Exponent 18could be replaced by 12. See also Rivera and Vanneuville (2017).

General result

Theorem (Abstract RSW)

Let fn

be a collection of Gaussian fields with covariance kernels n

.We assume that there are s

n

! 0 such that

1 Fields fn

and kernels n

are symmetric

2 Fields are smooth and non-degenerate

3 n

! K1 locally uniformly on scale sn

4 Asymptotically non-negative correlations

5 Uniform rapid decay of correlations

Then the nodal lines satisfy RSW on down to the scale sn

andnodal domains satisfy RSW on all scales.

Ingredients

For each field there is the minimal scale sn

. All estimatesshould be uniform

With high probability the nodal structure of a Gaussian field isdetermined by its discretization

If correlation function decays fast enough then crossings eventare asymptotically independent

Tassion’s argument could be adapted to the spherical setting

Small perturbations of the kernel change probability only alittle bit

Ingredients

For each field there is the minimal scale sn

. All estimatesshould be uniform

For Kostlan ensemble sn

=pn. On this scale it converges to

Bargmann-Fock

With high probability the nodal structure of a Gaussian field isdetermined by its discretization

If correlation function decays fast enough then crossings eventare asymptotically independent

Tassion’s argument could be adapted to the spherical setting

Small perturbations of the kernel change probability only alittle bit

Ingredients

For each field there is the minimal scale sn

. All estimatesshould be uniform

With high probability the nodal structure of a Gaussian field isdetermined by its discretization

If correlation function decays fast enough then crossings eventare asymptotically independent

Tassion’s argument could be adapted to the spherical setting

Small perturbations of the kernel change probability only alittle bit

Ingredients

For each field there is the minimal scale sn

. All estimatesshould be uniform

With high probability the nodal structure of a Gaussian field isdetermined by its discretization

If correlation function decays fast enough then crossings eventare asymptotically independent

Tassion’s argument could be adapted to the spherical setting

Small perturbations of the kernel change probability only alittle bit

Ingredients

For each field there is the minimal scale sn

. All estimatesshould be uniform

With high probability the nodal structure of a Gaussian field isdetermined by its discretization

If correlation function decays fast enough then crossings eventare asymptotically independent

For ✏ > 0 and polygons of controlled shape that are Csn

separatedfor large C

|P(A \ B)� P(A)P(B)| ✏

where A and B are crossing events

Tassion’s argument could be adapted to the spherical setting

Small perturbations of the kernel change probability only alittle bit

Ingredients

For each field there is the minimal scale sn

. All estimatesshould be uniform

With high probability the nodal structure of a Gaussian field isdetermined by its discretization

If correlation function decays fast enough then crossings eventare asymptotically independent

Tassion’s argument could be adapted to the spherical setting

Small perturbations of the kernel change probability only alittle bit

Ingredients

For each field there is the minimal scale sn

. All estimatesshould be uniform

With high probability the nodal structure of a Gaussian field isdetermined by its discretization

If correlation function decays fast enough then crossings eventare asymptotically independent

Tassion’s argument could be adapted to the spherical setting

Problems: no scaling, curvature

Small perturbations of the kernel change probability only alittle bit

Ingredients

For each field there is the minimal scale sn

. All estimatesshould be uniformWith high probability the nodal structure of a Gaussian field isdetermined by its discretizationIf correlation function decays fast enough then crossings eventare asymptotically independentTassion’s argument could be adapted to the spherical setting

Small perturbations of the kernel change probability only alittle bit

Ingredients

For each field there is the minimal scale sn

. All estimatesshould be uniformWith high probability the nodal structure of a Gaussian field isdetermined by its discretizationIf correlation function decays fast enough then crossings eventare asymptotically independentTassion’s argument could be adapted to the spherical setting

Small perturbations of the kernel change probability only alittle bit

Ingredients

For each field there is the minimal scale sn

. All estimatesshould be uniform

With high probability the nodal structure of a Gaussian field isdetermined by its discretization

If correlation function decays fast enough then crossings eventare asymptotically independent

Tassion’s argument could be adapted to the spherical setting

Small perturbations of the kernel change probability only alittle bit

Ingredients

For each field there is the minimal scale sn

. All estimatesshould be uniform

With high probability the nodal structure of a Gaussian field isdetermined by its discretization

If correlation function decays fast enough then crossings eventare asymptotically independent

Tassion’s argument could be adapted to the spherical setting

Small perturbations of the kernel change probability only alittle bit

Can add a small positive constant to the kernel. This allows toignore su�ciently small negative correlations

Ingredients

For each field there is the minimal scale sn

. All estimatesshould be uniform

With high probability the nodal structure of a Gaussian field isdetermined by its discretization

If correlation function decays fast enough then crossings eventare asymptotically independent

Tassion’s argument could be adapted to the spherical setting

Small perturbations of the kernel change probability only alittle bit