Volume distribution of nodal domains of random band-limited - PowerPoint Presentation

Slide1

Volume distribution of nodal domains of random band-limited functions

Indian Institute of ScienceApril 27, 2017

Dmitry Beliaev, OxfordIgor Wigman, KCL

Slide2

1. Motivation & Background

Slide3

Chladni plates video

Slide4

General Setup

– Compact smooth n-manifold

(most interesting n=2)

Laplace-Beltrami on M Eigenfunctions

:

Orthonormal basis of L

2(M,dVol), (Weyl law)

 

Slide5

Nodal components & domains

Nodal set:

Nodal components:

Connected components of

.

Smooth (n-1)-manifold.

Nodal domains:

Connected components of

smooth n-manifold  

Slide6

Questions – local vs. nonlocal I. Local (“easier” via Kac

-Rice): a. Total volume, nodal intersections, curvature… b. Critical points (values) c. Euler characteristic (Gauss-Bonnet).II. Nonlocal (semilocal

): a. Nodal count (Nazarov-Sodin `09,`12, `15) b. Topology: genus, area…; Nesting (Sarnak-W)

c. Geometry: Lengths/areas/volumes d*. Total Betti Number (Gayet-Welschinger

)

*

semilocality

not established

Slide7

“Semilocality” (Nazarov-Sodin)

Locality: (example) If then

+

Semilocality

:

“Most” of the nodal domains lie in

, R>>0.

Few “big” nodal domains.May approximate the total number of domains locally. 

Slide8

Nodal count (deterministic)

Nodal Count. Courant:

Pleijel:

Constant improved

by

(

Bourgain

)

No lower bound

 Nodal picture for the square, arbitrarily high energy. A. Stern’s thesis, Gottingen, 1925.Courtesy of P. Sarnak.

Slide9

Berry’s Random Wave Model

Principle (2d): as

“behave randomly” as wavenumber- monochromatic wave on

(chaotic M)

Random directions

, phases

.

Scale invariant, assume

Centered Gaussian, covariance

 

Slide10

2. Random Band Limited Functions

Slide11

Random Band-Limited FunctionsFix M – smooth n-manifold,

1

,

- N(0,1)

i.i.d

.

(

summation over

)

Covariance function

i.e. the spectral projector

.

 

Slide12

Limiting ensemblesNatural scaling around any point of M.

Scaling for covariance (values & derivatives)

Fix x, chart around x,

r

escale

Define

on

, “clean” covariance

Spectral measure characteristic on annulus

 

Slide13

Limiting ensembles (cont.)Defined Gaussian random field

on

(“scale invariant”).

is scaling limit of

around each point

depends on

,

not

on M, x (universality)Relevant question: nodal structures of restricted on ball , E.g. nodal count of domains lying in . 

Slide14

1. Random Spherical harmonics

2-dimensional unit sphere.

Eigenfunctions – spherical harmonics degree

. More precisely

,

Take

be orthonormal basis

,

are N(0,1)

i.i.d

.

Legendre

E.g.

Berry’s RWM

 

Slide15

Spherical harmonics vs. Planar waves

Random spherical

harmonicsAlex BarnettBerry’s RWM

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2. “Real Fubini Study”

n-dimensional projective space, round metric

– degree t homogeneous polynomial degree

,

is “random projective

hypesurface

”

 

Slide17

Some pictures (Alex Barnett)

=0

Real

Fubini-Study

 

Random spherical harmonics

 

Berry’s Random Waves

Slide18

Nodal count (Nazarov-Sodin)

Expected nodal count: (2007, 2012)

Concentration

(2012)

Exponential concentration

(2007)

,

degree

spherical harmonic

Valid for

toral

eigenfunctions

(

Rozenshein

`16).

Open

in general.

 

Slide19

3. Universal law for nodal volume distribution(Dmitry

Beliaev-IW `16)

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Scale-invariant (Euclidean) case Field e.g.

RWM on

For

,

let

number nodal domains

of

: a.

b. - number domains in

Nazarov-Sodin

: exists

s.t.

. Convergence in mean

, also

a.s

. (“N-S constant”)

Gaussian fields

(axioms), include

 

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Scale-invariant (Euclidean) case

Faber-Krahn:

Theorem (

Beliaev

-W): For every

there exists

s.t.

(a.s.) Denote

deterministic

increasing on

(optimal)

distribution function.

Caveat: continuous (differentiable) outside

countable

atoms.

Variety of Gaussian fields on

(

axiomatize

)

 

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Random spherical harmonics

Inherit

,

countable

Need to rescale area compared to

:

:

Theorem (

Beliaev

-W):

We have

for

The mass around atoms might be spreading

 

Slide23

Discussion In practice: for “every” instance of

the proportion of nodal domains with area

is about

deterministic. Support of area distribution (optimal):

Proof for

inspired

Canzani-Sarnak `16 Conjecture: 1. No “distinguished areas” 2. is everywhere differentiable on 3. on

 

Slide24

Coupling area + boundary length Can count the number of domains

of

Deterministic limit distribution

(

) in support must satisfy

isoperimetric inequality (“Dido’s Problem”). Eigenfunctions,

strictly increasing

,

(“optimal”). Proof inspired

Canzani-Sarnak

.

“Both oval-line and sausage-like domains exist”.

 

Slide25

Dido Building Cartphage / J. M. W. TurnerThe National Gallery

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The Embarkation of the Queen of Sheba / Claude LorrainThe National Gallery

Slide27

General CaseBand-limited

Euclidean

.

domains

;

,

limit law,

atoms

For

:

,

then

increasing on

 

Slide28

General case (cont.)

,

increasing on

1

st

zero of

Proof inspired by

Canzani-Sarnak

1. No “distinguished

volumes”

2

.

is

differentiable where increasing

3

.

domain of increase