Variation of the Nazarov-Sodin constant for random plane waves and - - PowerPoint PPT Presentation

Variation of the Nazarov-Sodin constant for random plane waves and arithmetic random waves

Random Waves in Oxford

Oxford, June 19, 2018

Par Kurlberg (KTH Stockholm) Igor Wigman, KCL

• 1. Motivation

& Background

General Setup



(𝑁, 𝑕) – Compact smooth surface (can generalize higher dimensions)



Δ Laplace-Beltrami on M



Eigenfunctions: (boundary condition) 𝜇𝑘 ≥ 0 Δ𝜒𝑘 + 𝜇𝑘𝜒𝑘 = 0

 Orthonormal basis of L2(M,dVol), 𝜇𝑘 → ∞

Nodal components & domains

 Nodal set: 𝑎 𝜒𝑘 = 𝜒𝑘

−1(0)

 Nodal components: Connected components

• f 𝜒𝑘

−1(0).

 Nodal domains: Connected components of

𝑁 ∖ 𝜒𝑘

−1(0) smooth

 Nodal count: How many

components (domains)?

Interesting Questions – non-local



Nodal count (Nazarov-Sodin `09,`12, `15, Kurlberg-W `14, `17)

 T

• pology, nesting,geometry(Sarnak-W, Beliaev-W)

 Local: 𝑀𝑓𝑜𝐵∪𝐶 𝑔

= 𝑀𝑓𝑜𝐵 𝑔 + 𝑀𝑓𝑜𝐶 𝑔 𝐵 ∩ 𝐶 = ∅

 Semilocality: “Most” of the nodal domains of

diameter

𝑆 𝜇, R>>0.

 Approximate locally

Nodal count (deterministic)

 Nodal Count. Courant: 𝑂 𝜒𝑘 ≤ 𝑘  Pleijel: limsup

𝑘→∞

𝑂 𝜒𝑘

𝑘

≤ 0.691 …

 Constant improved by 3 ∙ 10−9 (Bourgain)  No lower bound 𝑂 𝜒𝑘 ≥ 2

Nodal picture for the square, arbitrarily high

• energy. A. Stern’s thesis,

Gottingen, 1925. Courtesy of P. Sarnak.

Berry’s Random Wave Model

 M chaotic. As 𝜇 → ∞, 𝜒𝑘 “behave randomly”

wavenumber 𝜇 monochromatic wave ℝ2 𝑣 𝜇(𝑦) =

1 𝐾 𝔒𝑓 𝑘=1 𝐾

𝑓𝑗( 𝜇 𝑦,𝜘𝑘 +𝜔𝑘)

 Scale invariant, assume 𝑣 = 𝑣1  Centered Gaussian, covariance

E[𝑣(𝑦) ∙ 𝑣(𝑧)] = 𝐾0(|𝑦 − 𝑧|)

 Spectral measure – arc length on unit circle

• 2. Random Band

Limited Functions

Random Band-Limited Functions

 Fix M – smooth n-manifold, 0 ≤ 𝛽 ≤1

𝑔

𝜇 x = 𝛽𝜇≤𝜇𝑘≤𝜇 𝑏𝑘𝜒𝑘(𝑦), 𝑏𝑘 - N(0,1) i.i.d.

(𝛽 = 1 summation over 𝜇 − 𝑝 𝜇 ≤ 𝜇𝑘 ≤ 𝜇)

 Covariance function

𝔽 𝑔

𝜇 𝑦 𝑔 𝜇 𝑧

= 𝜒𝑘(𝑦)𝜒𝑘(𝑧) i.e. the spectral projector.

Example 1. Random Spherical harmonics

 𝛽 = 1, 𝑁 = 𝒯2, 2d sphere.  𝔽 𝑈𝑚 x ∙ 𝑈𝑚 y

= 𝑄𝑚 cos(𝑒 𝑦, 𝑧 ) .

 𝑄𝑚 cos(𝑒) ≈ 𝐾0 𝑚𝑒 Legendre fast uniform  Scales Berry’s RWM

Random spherical harmonics

• A. Barnett

RWM

𝛽=1 vs 𝛽=0 (Alex Barnett)

𝛽=0 “Real Fubini-Study” 𝛽=1 Random spherical harmonics

Example 2. T

• ral eigenfunctions.

 𝕌 = 𝑁 =

ℝ2 ℤ2

 𝑔

𝑜 𝑦 = 𝜈 2=𝑜 𝑏𝜈 ∙ 𝑓

𝑦, 𝜈 𝑏𝜈 standard Gaussian i.i.d. (save to 𝑏−𝜈 = 𝑏𝜈) “arithmetic random waves”

 Summation over 𝜈 ∈ ℤ2:

𝜈 2 = 𝑜 lattice points on radius 𝑜 circle

More general: limiting ensembles

 Natural scaling around any point of M.  Scaling for covariance (values & derivatives)

(classical Hormander, Lax) 𝔽 𝑔

𝜇 x ∙ 𝑔 𝜇 y

≈ 𝐿𝛽 𝜇 ∙ 𝑒 𝑦, 𝑧

 𝐿𝛽 𝑥 = 𝐿𝛽

𝑥 =

𝛽≤ 𝑥 ≤1 𝑓 𝑥, 𝜊

𝑒𝜊

 Canzani-Hanin `16 𝛽 = 1 thin window.  Define 𝑕∞on ℝ2, “clean” covariance

𝔽[𝑕∞ 𝑨 ∙ 𝑕∞ 𝑨′ ] = 𝐿𝛽 𝑨 − 𝑨′

Limiting ensembles (cont.)

 𝔽[𝑕∞ 𝑨 ∙ 𝑕∞ 𝑨′ ] = 𝐿𝛽

𝑨 − 𝑨′

 𝑕∞ scaling limit 𝑔

𝜇 (everywhere)

 𝑕∞ depends on 𝛽, not on M, x (universality)  Spectral measure  Relevant: nodal structures of 𝑕∞ restricted on

ball 𝐶 𝑆 , 𝑆 → ∞.

 E.g. nodal count of domains lying in 𝐶 𝑆 .

𝛽 1

• 3. Nazarov-Sodin

Constant

Scale invariant (Euclidean) case

 𝐺: ℝ2 → ℝ stationary Gaussian field  𝜍 spectral measure of 𝐺  𝑂(𝐺; 𝑆) is the number of connected

components (domains) of 𝐺 inside 𝐶(𝑆)

 Assuming: 1. 𝐺 ergodic (𝜍 has no atoms)

• 2. 𝐺 smooth. 3. Non-degeneracy

 Nazarov-Sodin (`12,`15): 𝑑 = 𝑑𝑂𝑇(𝜍) ≥ 0

𝐹[𝑂 𝐺; 𝑆 ] = 𝑑 ∙ 𝑆2 + 𝑝𝑆→∞(𝑆2)

 “Usually” 𝑑 > 0 (support of 𝜍)

Nodal count band-limited functions

 𝑑 = 𝑑𝑂𝑇(𝜍), E 𝑂 𝐺; 𝑆

= 𝑑 ∙ 𝑆2 + 𝑝𝑆→∞(𝑆2)

 Stronger convergence in mean (ergodicity)

E

𝑂 𝐺;𝑆 −𝑑∙𝑆2 𝑆2

→ 0

 Band-limited functions 𝑑 = 𝑑𝑂𝑇 𝜍𝛽 > 0

𝜍𝛽 area measure annulus

 E[𝑂(𝑔

𝜇)]~𝑑 ∙ 𝜇 (NS `12,`15)

 Convergence in mean

1

𝛽

Random spherical harmonics

 𝑣 =Plane monochromatic waves (RWM)  𝐹[𝑂 𝑣; 𝑆 ]~𝑑𝑆𝑋𝑁 ∙ 𝑆2, universal NS constant  𝑑𝑆𝑋𝑁 = 𝑑𝑂𝑇

𝑒𝜄 2𝜌 > 0 percolation?

 𝐹 𝑂(𝑈𝑚) ~𝑑𝑆𝑋𝑁 ∙ 𝑚2 (NS `09)  Convergence in mean  Exponential probability concentration

1

• 4. Variation on

Nazarov-Sodin consant

Generalise NS constant

 Restrict to 𝜍 supported on the unit ball 𝒬

(spectral moments), includes band limited case

 Proposition 1(Kurlberg-W): c = 𝑑𝑂𝑇(𝜍),

E[𝑂 𝐺; 𝑆 ] = 𝑑 ∙ 𝑆2 + 𝑃(𝑆), 𝜍 ∈ 𝒬 arbitrary, absolute constant (uniform)

 𝑑𝑂𝑇(𝜍) bounded (e.g. critical points Kac-Rice)  No convergence in mean. Can construct

examples E

𝑂 𝐺;𝑆 −𝑑∙𝑆2 𝑆2

doesn’t vanish

 Example: 𝜍 atomic supported at 0. 𝐺 ≡ 𝑑𝑝𝑜𝑡𝑢,

Gaussian 𝑂 𝐺; 𝑆 ≡ 0, ⇒ 𝑑𝑂𝑇 𝜍 = 0.

Variation of NS constant

 Proposition 2 (Kurlberg-W):

𝑒𝑂𝑇 𝜍 ≔ lim

𝑆→∞ E 𝑂 𝐺;𝑆 −𝑑∙𝑆2 𝑆2

exists (“NS discrepancy functional”) non-uniform, discontinous

 Theorem 1 (Kurlberg-W): 𝑑𝑂𝑇 𝜍 : 𝒬 → ℝ≥0 is

continuous (weak* topology on 𝒬).

 Corollary: 𝑑𝑂𝑇 𝜍 attains an interval [0, 𝑑0]

(𝒬 is essential)

 Q: Is it true that 𝑑 = 𝑑𝑆𝑋𝑁, uniquely?

• 5. Toral Eigenfunctions

Example 2. T

• ral eigenfunctions.

 𝕌 = 𝑁 =

ℝ2 ℤ2

 𝑔

𝑜 𝑦 = 𝜈 2=𝑜 𝑏𝜈 ∙ 𝑓

𝑦, 𝜈 𝑏𝜈 standard Gaussian i.i.d. (save to 𝑏−𝜈 = 𝑏𝜈) “arithmetic random waves”

 Summation over 𝜈 ∈ ℤ2:

𝜈 2 = 𝑜 lattice points on radius 𝑜 circle

 𝑂(𝑔

𝑜 𝑦 ) nodal count

On the 2 squares problem

 𝑠2 𝑜 = #

𝑏, 𝑐 𝜗ℤ2: 𝑏2 + 𝑐2 = 𝑜

 On average 𝑠2 𝑜 ~c ∙

log(𝑜) (E. Landau) Equidistributed generic

 Partial classification (P

. Kurlberg-IW `15)

1

1

𝑠2(n) → ∞ Exceptional “Cilleruelo”

Some pics

𝑜 = 1105 32 directions 𝑜 = 9676418088513347624474653 256 directions Fragment, domains long and narrow

On the 2 squares problem

 𝜐𝑜 =

1 𝑠2(𝑜) 𝜈 2=𝑜 𝜀𝜈/ 𝑜 probability 𝒯1

(spectral measure)

 Equidistributed

𝜐𝑜𝑘 ⇒

𝑒𝜄 2𝜌

(𝜐𝑜𝑘 ⇒ 𝜐)

 Angular distribution ↭ Nodal structure

Local: Krishnapur-Kurlberg-W `13, Rudnick-W `14, Rossi-W `17

 Nonlocal: N(𝑔

𝑜 𝑦 ) – total nodal count

Nazarov-Sodin `12,`15+Kurlberg-W `14,17

T

• ral eigenfunctions

 𝑔

𝑜 𝑦 = 𝜈 2=𝑜 𝑏𝜈 ∙ 𝑓 𝑦, 𝜈

arithmetic random waves

 𝜐𝑜 =

1 𝑠2(𝑜) 𝜈 2=𝑜 𝜀𝜈/ 𝑜 on 𝑇1

 Apply N-S: if 𝜐𝑜 ⇒ 𝜐 then 𝑑 = 𝑑𝑂𝑇 𝜐

(generalised) 𝐹 𝑂(𝑔

𝑜 𝑦 ) = 𝑑 ∙ 𝑜 + 𝑝 𝑜

 Generic 𝐹

𝑂 𝑔

𝑜 𝑦

−𝑑𝑆𝑋𝑁 𝑜

→ 0

 Exponential concentration (Y. Rozenshein `15)

T

• ral eigenfunctions (cont.)

 𝑔

𝑜 𝑦 = 𝜈 2=𝑜 𝑏𝜈 ∙ 𝑓 𝑦, 𝜈

arithmetic random waves

 𝜐𝑜 =

1 𝑠2(𝑜) 𝜈 2=𝑜 𝜀𝜈/ 𝑜 on 𝑇1

 Theorem 2 (Kurlberg-W):

• 1. Uniformly

𝐹 𝑂(𝑔

𝑜 𝑦 ) = 𝑑𝑂𝑇(𝜐𝑜) ∙ 𝑜 + 𝑃

𝑜

• 2. 𝑑𝑂𝑇 𝜐 = 0 iff 𝜐 is Cilleruelo or its tilt

(𝜐 restricted, in particular by symmetries)

T

• ral eigenfunctions (cont.)

 𝑑𝑂𝑇 𝜐 = 0 iff 𝜐 is Cilleruelo or its tilt

(restricted)

 𝑑𝑂𝑇 𝜐 attains an interval 0, 𝑑1 .  Q.: Is it true that 𝑑1 = 𝑑0 = 𝑑𝑆𝑋𝑁 uniquely?  Q.: For Cilleruelo: 𝐹 𝑂(𝑔

𝑜 𝑦 ) - ?

𝐹 𝑂(𝑔

𝑜 𝑦 ) → ∞ - ?

𝐹 𝑂(𝑔

𝑜 𝑦 ) ≫

𝑜 - ?

 Meanwhile full classification 𝑑𝑂𝑇 𝜍 = 0

(Beliaev-McAuley-Muirhead). Same for 𝑒𝑂𝑇 𝜍 ?