On the spatial distribution of critical points of Random Plane Waves

Transcription

1 On the spatial distribution of critical points of Random Plane Waves Valentina Cammarota Department of Mathematics, King s College London Workshop on Probabilistic Methods in Spectral Geometry and PDE CRM Montreal August

2 Random Plane Wave A 1-dim version of the Random Wave was used by Rice to investigate the likelihood of a given signal to exceed a level. In 1950 Longuet-Higgins generalised Rice s model to 2-dim plane to describe ocean waves. In 1977 M. Berry conjectured that high energy behaviour of eigenfunctions in the chaotic case is universal and have statistically the same behaviour as random plane wave.

3 Random Plane Wave Space of spherical harmonics of degree l is of dimension 2l + 1, let {φ i } i=1,...2l+1 be an arbitrary L 2 -orthonormal basis. Define random Gaussian spherical harmonic f l = 2l+1 i=1 c i φ i c i are i.i.d. standard Gaussian variables. RPW is the scaling limit of the Gaussian spherical harmonic. We can relate high energy Gaussian spherical harmonics with to the the behaviour of RPW inside a big ball. Here we focus on critical points in small balls and repulsion.

4 Definition Random Plane Wave The Random Plane Wave with energy E = k 2 is the centred Gaussian field on R 2 with covariance kernel K(x, y) = J 0 (k x y ). One may think of RPW as a random Gaussian solution of f + k 2 f = 0 i.e. F (x) = c n J n (kr)e inθ n= c n are standard Gaussian independent save to c n = c n.

5 Critical Points We study the critical point set {x R 2 : F (x) = 0}. We are interested in the spatial distribution of critical points across the surface. We fix k = 1 since RPW with different values of k differ by the scaling.

6 Nodal lines {x R 2 : F (x) = 0} Nodal domains R 2 \{x R 2 : F (x) = 0} Motivations Nodal domains for k = 100. Two nodal domains are highlighted [Bogomolny- Schmit 2002]

7 Motivations [Bogomolny-Schmit 2002] introduce a bond percolation model on the square lattice to describe nodal domains.

8 Motivations [Beliaev-Kereta 2013] introduce a bond percolation on a random graph where nodes are local maxima and edges are gradient streamlines passing through saddles. Pictures by D. Beliaev and T. Sharpe.

9 Critical points of RPW. Picture by D. Beliaev. 1 Motivations

10 Motivations 2 1. IN Figure : Critical points Repulsion? Figure : Poisson FIGURE 1. Samples of trans plane: Poisson (left), determ K(z, w) = 1 º ezw 1 2 ( z 2 + w 2). D sion, while permanental proc one of two important classes of point p study in this book.

11 Expectation and Variance N c ρ = #{x B(ρ) : F (x) = 0} Theorem (Beliaev-C.-Wigman) For every ρ > 0 As ρ 0 E[N c ρ (N c ρ 1)] = E[N c ρ ] = ρ ρ4 + O(ρ 6 ).

12 Proof Critical points density K 1 : R 2 R K 1 (x) = φ F (x) (0, 0) E[ deth F (x) F (x) = 0] Two-point correlation function K 2 : R 2 R 2 R K 2 (x, y) = φ ( F (x), F (y)) (0, 0) E[ deth F (x) deth F (y) F (x) = F (y) = 0].

13 Proof We apply a suitably modified version of Kac-Rice formula for counting the number of zeros of the gradient of F : and, for x y, E[Nρ c ] = E[Nρ c (Nρ c 1)] = B(0,ρ) K 1 (x)dx, B(0,ρ) B(0,ρ) K 2 (x, y)dxdy; provided that the Gaussian distribution of ( F (x), F (y)) R 4 is non-degenerate for all (x, y) B(0, ρ) B(0, ρ).

14 Proof Lemma For every x R 2 K 1 (x) = 1 2 3π. As r = d(x, y) 0 K 2 (r) = π 2 + O(r2 ).

15 No repulsion! 2 1. IN As r 0 Figure : Critical points K 2 (r) (K 1 (r)) 2 = O(r2 ) r > 0 Figure : Poisson FIGURE 1. Samples of trans plane: Poisson (left), determ K(z, w) = 1 º ezw 1 2 ( z 2 + w 2). D sion, while permanental proc K 2 (r) one of two (Kimportant 1 (r)) 2 = 1 classes of point p study in this book.

16 Probability of 0, 1 pts N c ρ = #{x B(ρ) : F (x) = 0} Corollary As ρ 0 P(N c ρ = 0) = ρ2 + O(ρ 4 ), P(N c ρ = 1) = ρ2 + O(ρ 4 ).

17 P(N c ρ = 1) P(N c ρ 1) We note that k=1 E[N ] = P(N = 1) + k=2 Proof k P(N c ρ = k) = E[N c ρ ] = ρ2. k P(N = k) and P(N = 1) = E[N ] k P(N = k) E[N ] k(k 1) P(N = k) then = E[N ] E[N (N 1)] k=2 k=2 P(Nρ c = 1) ρ ρ4 + O(ρ 6 ).

18 Proof - Lemma Critical point density K 1 K 1 (x) = φ F (x) (0, 0) E[ deth F (x) F (x) = 0], where and φ F (x) (0, 0) = 1 π, E[ deth F (x) F (x) = 0] = E[ deth F (x) ] = Two-point correlation function K 2 around the origin = perturbation theory.

19 Proof - Lemma Write the two-point correlation K 2 function as a function of the perturbing elements of the covariance matrix (r) of ( 2 F (x), 2 F (y) F (x) = F (y) = 0). The Gaussian expectations K 2 is analytic functions of the of the perturbing elements of (r). K 2 is a smooth function, defined on some neighbourhood of the origin. We can Taylor expand K 2 around the origin.

20 Proof - Lemma 1 K 2 (r) = (2π) 5 x 1 x 3 x 2 2 x 4 x 6 x 2 5 deta(r) R { 6 1 exp 1 } det (r) 2 xt (r) 1 x dx. For every r > 0 (r) is symmetric = we diagonalise (r): (r) = P (r) t Λ(r)P (r) Compute the eigenvalues and eigenvectors of the perturbed matrix (r). Note that 1 exp{ 1 det (r) 2 xt (r) 1 1 x} = exp{ 1 λi(r) 2 (P (r)x)t Λ(r) 1 P (r)x}. Change variable: z = P (r)x i.e. x = P (r) 1 z.

21 Proof - Lemma Finally K 2 (r) { } 1 = f(r, z) exp (2π) zi 2 dz deta(r) R 2 6 i=1 1 [ r 2 { } = π 5 2 z 2 3r 2 + O(r 4 ) 384 4z6 2 exp 1 6 ] zi 2 dz + O(r 4 ) R 2 6 = π 2 + O(r2 ). i=1

22 E[N c ρ (N c ρ 1) (N c ρ 2)] = O(ρ 6 ) And... = P(N c ρ = 2) = ρ4 + O(ρ 6 ) Repulsion between saddles?

23 Thank You!