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 Par Kurlberg (KTH Stockholm) Igor Wigman, KCL Random Waves in Oxford Oxford, June 19, 2018

1. Motivation & Background

General Setup (𝑁, 𝑕) – Compact smooth surface (can  generalize higher dimensions) Δ Laplace-Beltrami on M  Eigenfunctions: (boundary condition)  𝜇 𝑘 ≥ 0 Δ𝜒 𝑘 + 𝜇 𝑘 𝜒 𝑘 = 0  Orthonormal basis of L 2 (M,dVol), 𝜇 𝑘 → ∞

Nodal components & domains −1 (0)  Nodal set: 𝑎 𝜒 𝑘 = 𝜒 𝑘  Nodal components: Connected components −1 (0) . of 𝜒 𝑘  Nodal domains: Connected components of −1 (0) smooth 𝑁 ∖ 𝜒 𝑘  Nodal count: How many components (domains)?

Interesting Questions – non-local Nodal count (Nazarov-Sodin `09,`12, `15,  Kurlberg-W `14, `17)  T opology, nesting,geometry(Sarnak-W, Beliaev-W)  Local: 𝑀𝑓𝑜 𝐵∪𝐶 𝑔 = 𝑀𝑓𝑜 𝐵 𝑔 + 𝑀𝑓𝑜 𝐶 𝑔 𝐵 ∩ 𝐶 = ∅  Semilocality: “Most” of the nodal domains of 𝑆 diameter 𝜇 , R>>0.  Approximate locally

Nodal count (deterministic)  Nodal Count. Courant: 𝑂 𝜒 𝑘 ≤ 𝑘 𝑂 𝜒 𝑘  Pleijel: limsup ≤ 0.691 … 𝑘 𝑘→∞  Constant improved by 3 ∙ 10 −9 (Bourgain)  No lower bound 𝑂 𝜒 𝑘 ≥ 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 → ∞, 𝜒 𝑘 “behave randomly” wavenumber 𝜇 monochromatic wave ℝ 2 1 𝐾 𝑓 𝑗( 𝜇 𝑦,𝜘 𝑘 +𝜔 𝑘 ) 𝐾 𝔒𝑓 𝑘=1 𝑣 𝜇 (𝑦) =  Scale invariant, assume 𝑣 = 𝑣 1  Centered Gaussian, covariance E[𝑣(𝑦) ∙ 𝑣(𝑧)] = 𝐾 0 (|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 𝜇 x = 𝛽𝜇≤𝜇 𝑘 ≤𝜇 𝑏 𝑘 𝜒 𝑘 (𝑦) , 𝑏 𝑘 - N(0,1) i.i.d. 𝑔 ( 𝛽 = 1 summation over 𝜇 − 𝑝 𝜇 ≤ 𝜇 𝑘 ≤ 𝜇 )  Covariance function 𝔽 𝑔 𝜇 𝑦 𝑔 𝜇 𝑧 = 𝜒 𝑘 (𝑦)𝜒 𝑘 (𝑧) i.e. the spectral projector.

Example 1. Random Spherical harmonics  𝛽 = 1, 𝑁 = 𝒯 2 , 2d sphere.  𝔽 𝑈 𝑚 x ∙ 𝑈 𝑚 y = 𝑄 𝑚 cos(𝑒 𝑦, 𝑧 ) .  𝑄 𝑚 cos(𝑒) ≈ 𝐾 0 𝑚𝑒 Legendre fast uniform  Scales Berry’s RWM RWM Random spherical harmonics A. Barnett

𝛽 =1 vs 𝛽 =0 (Alex Barnett) 𝛽 =1 𝛽 =0 Random spherical harmonics “Real Fubini- Study”

Example 2. T oral eigenfunctions. ℝ 2  𝕌 = 𝑁 = ℤ 2 𝑜 𝑦 = 𝜈 2 =𝑜 𝑏 𝜈 ∙ 𝑓  𝑔 𝑦, 𝜈 𝑏 𝜈 standard Gaussian i.i.d. (save to 𝑏 −𝜈 = 𝑏 𝜈 ) “ arithmetic random waves ” 𝜈 2 = 𝑜  Summation over 𝜈 ∈ ℤ 2 : 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 ≈ 𝐿 𝛽 𝜇 ∙ 𝑒 𝑦, 𝑧  𝐿 𝛽 𝑥 = 𝐿 𝛽 𝑥 = 𝛽≤ 𝑥 ≤1 𝑓 𝑥, 𝜊 𝑒𝜊  Canzani-Hanin `16 𝛽 = 1 thin window.  Define 𝑕 ∞ on ℝ 2 , “clean” covariance 𝔽[𝑕 ∞ 𝑨 ∙ 𝑕 ∞ 𝑨′ ] = 𝐿 𝛽 𝑨 − 𝑨′

Limiting ensembles (cont.)  𝔽[𝑕 ∞ 𝑨 ∙ 𝑕 ∞ 𝑨′ ] = 𝐿 𝛽 𝑨 − 𝑨′  𝑕 ∞ scaling limit 𝑔 𝜇 (everywhere)  𝑕 ∞ depends on 𝛽 , not on M, x (universality)  Spectral measure 𝛽 1  Relevant: nodal structures of 𝑕 ∞ restricted on ball 𝐶 𝑆 , 𝑆 → ∞.  E.g. nodal count of domains lying in 𝐶 𝑆 .

3. Nazarov-Sodin Constant

Scale invariant (Euclidean) case  𝐺: ℝ 2 → 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 = 𝑑 𝑂𝑇 (𝜍) ≥ 0 𝐹[𝑂 𝐺; 𝑆 ] = 𝑑 ∙ 𝑆 2 + 𝑝 𝑆→∞ (𝑆 2 )  “Usually” 𝑑 > 0 (support of 𝜍 )

Nodal count band-limited functions = 𝑑 ∙ 𝑆 2 + 𝑝 𝑆→∞ (𝑆 2 )  𝑑 = 𝑑 𝑂𝑇 (𝜍) , E 𝑂 𝐺; 𝑆  Stronger convergence in mean (ergodicity) 𝑂 𝐺;𝑆 −𝑑∙𝑆 2 E → 0 𝑆 2  Band-limited functions 𝑑 = 𝑑 𝑂𝑇 𝜍 𝛽 > 0 𝜍 𝛽 area measure annulus 𝜇 )]~𝑑 ∙ 𝜇 (NS `12,`15)  E[𝑂(𝑔 𝛽 1  Convergence in mean

Random spherical harmonics  𝑣 = Plane monochromatic waves (RWM)  𝐹[𝑂 𝑣; 𝑆 ]~𝑑 𝑆𝑋𝑁 ∙ 𝑆 2 , universal NS constant 2𝜌 > 0 percolation? 𝑒𝜄  𝑑 𝑆𝑋𝑁 = 𝑑 𝑂𝑇  𝐹 𝑂(𝑈 𝑚 ) ~𝑑 𝑆𝑋𝑁 ∙ 𝑚 2 (NS `09)  Convergence in mean 1  Exponential probability concentration

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 = 𝑑 𝑂𝑇 (𝜍) , E[𝑂 𝐺; 𝑆 ] = 𝑑 ∙ 𝑆 2 + 𝑃(𝑆) , 𝜍 ∈ 𝒬 arbitrary, absolute constant (uniform)  𝑑 𝑂𝑇 (𝜍) bounded (e.g. critical points Kac-Rice)  No convergence in mean. Can construct 𝑂 𝐺;𝑆 −𝑑∙𝑆 2 examples E doesn’t vanish 𝑆 2  Example: 𝜍 atomic supported at 0. 𝐺 ≡ 𝑑𝑝𝑜𝑡𝑢 , Gaussian 𝑂 𝐺; 𝑆 ≡ 0 , ⇒ 𝑑 𝑂𝑇 𝜍 = 0.

Variation of NS constant  Proposition 2 (Kurlberg-W): 𝑂 𝐺;𝑆 −𝑑∙𝑆 2 exists 𝑒 𝑂𝑇 𝜍 ≔ lim 𝑆→∞ E 𝑆 2 (“NS discrepancy functional”) non -uniform, discontinous  Theorem 1 (Kurlberg-W): 𝑑 𝑂𝑇 𝜍 : 𝒬 → ℝ ≥0 is continuous (weak* topology on 𝒬 ).  Corollary: 𝑑 𝑂𝑇 𝜍 attains an interval [0, 𝑑 0 ] ( 𝒬 is essential)  Q: Is it true that 𝑑 = 𝑑 𝑆𝑋𝑁 , uniquely ?

5. Toral Eigenfunctions

Example 2. T oral eigenfunctions. ℝ 2  𝕌 = 𝑁 = ℤ 2 𝑜 𝑦 = 𝜈 2 =𝑜 𝑏 𝜈 ∙ 𝑓  𝑔 𝑦, 𝜈 𝑏 𝜈 standard Gaussian i.i.d. (save to 𝑏 −𝜈 = 𝑏 𝜈 ) “ arithmetic random waves ” 𝜈 2 = 𝑜  Summation over 𝜈 ∈ ℤ 2 : lattice points on radius 𝑜 circle 𝑜 𝑦 ) nodal count  𝑂(𝑔

On the 2 squares problem 𝑏, 𝑐 𝜗ℤ 2 : 𝑏 2 + 𝑐 2 = 𝑜  𝑠 2 𝑜 = #  On average 𝑠 2 𝑜 ~c ∙ log(𝑜) (E. Landau) Equidistributed Exceptional “ Cilleruelo ” generic 𝑠 2 (n) → ∞ 1 1  Partial classification (P . Kurlberg-IW `15)

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

On the 2 squares problem 1 𝑠 2 (𝑜) 𝜈 2 =𝑜 𝜀 𝜈/ 𝑜 probability 𝒯 1  𝜐 𝑜 = (spectral measure) 𝑒𝜄  Equidistributed ( 𝜐 𝑜 𝑘 ⇒ 𝜐 ) 𝜐 𝑜 𝑘 ⇒ 2𝜌  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 oral eigenfunctions arithmetic 𝑜 𝑦 = 𝜈 2 =𝑜 𝑏 𝜈 ∙ 𝑓 𝑦, 𝜈  𝑔 random waves 1 𝑠 2 (𝑜) 𝜈 2 =𝑜 𝜀 𝜈/ 𝑜 on 𝑇 1  𝜐 𝑜 =  Apply N-S: if 𝜐 𝑜 ⇒ 𝜐 then 𝑑 = 𝑑 𝑂𝑇 𝜐 (generalised) 𝐹 𝑂(𝑔 𝑜 𝑦 ) = 𝑑 ∙ 𝑜 + 𝑝 𝑜 𝑂 𝑔 𝑜 𝑦 −𝑑 𝑆𝑋𝑁  Generic 𝐹 → 0 𝑜  Exponential concentration (Y. Rozenshein `15)

T oral eigenfunctions (cont.) arithmetic 𝑜 𝑦 = 𝜈 2 =𝑜 𝑏 𝜈 ∙ 𝑓 𝑦, 𝜈  𝑔 random waves 1 𝑠 2 (𝑜) 𝜈 2 =𝑜 𝜀 𝜈/ 𝑜 on 𝑇 1  𝜐 𝑜 =  Theorem 2 (Kurlberg-W): 1. Uniformly 𝐹 𝑂(𝑔 𝑜 𝑦 ) = 𝑑 𝑂𝑇 (𝜐 𝑜 ) ∙ 𝑜 + 𝑃 𝑜 2. 𝑑 𝑂𝑇 𝜐 = 0 iff 𝜐 is Cilleruelo or its tilt ( 𝜐 restricted, in particular by symmetries)

T oral eigenfunctions (cont.)  𝑑 𝑂𝑇 𝜐 = 0 iff 𝜐 is Cilleruelo or its tilt (restricted)  𝑑 𝑂𝑇 𝜐 attains an interval 0, 𝑑 1 .  Q.: Is it true that 𝑑 1 = 𝑑 0 = 𝑑 𝑆𝑋𝑁 uniquely?  Q.: For Cilleruelo: 𝐹 𝑂(𝑔 𝑜 𝑦 ) - ? 𝑜 𝑦 ) → ∞ - ? 𝐹 𝑂(𝑔 𝑜 - ? 𝐹 𝑂(𝑔 𝑜 𝑦 ) ≫  Meanwhile full classification 𝑑 𝑂𝑇 𝜍 = 0 (Beliaev-McAuley-Muirhead). Same for 𝑒 𝑂𝑇 𝜍 ?