III. Quantum ergodicity on graphs, perspectives

Transcription

1 III. Quantum ergodicity on graphs, perspectives Nalini Anantharaman Université de Strasbourg 24 août 2016

2 Yesterday we focussed on the case of large regular (discrete) graphs. Let G = (V, E) be a (q + 1)-regular graph. Discrete laplacian : f : V C, f (x) = (f (y) f (x)) = y x y x f (y) (q + 1)f (x). = A (q + 1)I

3 Sp(A) [ (q + 1), q + 1] Let V = N. We look at the limit N +.

4 Sp(A) [ (q + 1), q + 1] Let V = N. We look at the limit N +. We assume that G N has few short loops (= converges to a tree in the sense of Benjamini-Schramm).

5 Theorem (A-Le Masson, 2013) Assume that G N has few short loops and that it forms an expander family = uniform spectral gap for A. Let (φ (N) i ) N i=1 be an ONB of eigenfunctions of the laplacian on G N. Let a = a N : V N C be such that a(x) 1 for all x V N. Then lim N + 1 N N a(x) φ (N) i (x) 2 a x V N i=1 a = 1 a(x) N x V N 2 = 0.

6 Deterministic statement ; applies in particular to random regular graphs

7 More general version Theorem (A-Le Masson, 2013) Assume that G N has few short loops and that it forms an expander family. Let (φ (N) i ) N i=1 be an ONB of eigenfunctions of the laplacian on G N. Let K = K N : V N V N C be a matrix such that d(x, y) > D = K(x, y) = 0. Assume K(x, y) 1. Then lim N + 1 N N φ (N) i i=1, Kφ (N) i K λi 2 = 0.

8 Yesterday s proof yielded the following expression for K λ : thanks to Fourier analysis on the (q + 1)-regular tree, we wrote K = Op(a) and we obtained K λ = 1 N x D N X if λ = q 1/2+is + q 1/2 is = 2 q cos(s ln q). a(x, ω, s)dν x (ω)

9 Another formula for K λ K λ = 1 N x D,y X K(x, y)φ sph,λ (d(x, y)). Φ sph,λ is the spherical function of parameter λ on the (q + 1)-regular tree.

10 Another formula for K λ K λ = 1 N x D,y X K(x, y)φ sph,λ (d(x, y)). Φ sph,λ is the spherical function of parameter λ on the (q + 1)-regular tree. Φ λ (d) = q d/2 ( 2 q + 1 if λ = 2 q cos(s ln q). cos(ds ln q) + q 1 q + 1 ) sin((d + 1)s ln q) sin(s ln q)

11 A third formula for K λ Write G N = G = (V, E) = Γ\X. Introduce the Hilbert space H Γ = {(K(x, y)) (x,y) X X, K(γ x, γ y) = K(x, y) γ Γ, 1 K(x, y) 2 < + } D Consider the closed subspace x D,y X F = Vect{A k, k 0}={K H Γ, [A, K] = 0} H Γ.

12 A third formula for K λ For K H Γ, P F (K) = f K (A) where f K is a function on R (actually a polynomial of degree D if [d(x, y) > D = K(x, y) = 0])

13 A third formula for K λ For K H Γ, P F (K) = f K (A) where f K is a function on R (actually a polynomial of degree D if [d(x, y) > D = K(x, y) = 0]) We find K λ = f K (λ).

14 4 line (sketch of) proof Var([A, K]) = 0 Var(K) = 1 V N φ (N) i i=1 Var(K) 1 V K 2 HS(l 2 (V )) K 2 H ΓN, Kφ (N) i 2 + C(D) sup K 2 1 {x V, ρ(x) < D}. V By a density argument (using expansion), Var(K) 0 N + if K HΓN F. For general K, apply the previous line to K P F (K) = K f K (A).

15 On a regular graph, consider a weighted adjacency matrix, A p f (x) = y x p(x, y)f (y) with homogeneous probability weights.

16 Theorem (2015) Assume that G N has few short loops and that it forms an expander family. Let (φ (N) i ) N i=1 be an ONB of eigenfunctions of A p on G N. Let K = K N : V N V N C be a matrix such that d(x, y) > D = K(x, y) = 0. Assume K(x, y) 1. Then lim N + 1 N N φ (N) i i=1, Kφ (N) i K λi 2 = 0.

17 K λ = 1 N x,y K(x, y) Im G λ+i0(x, y) Im G λ+i0 (x, x). (the Green function of the infinite (q + 1)-regular tree)

18 K λ = 1 N x,y K(x, y) Im G λ+i0(x, y) Im G λ+i0 (x, x). (the Green function of the infinite (q + 1)-regular tree) Write the spectral decomposition of A p on X, dp (,λ] = P λ dm(λ) K λ = 1 V Tr l 2 (X)1l D KP λ.

19 Exploration + V (x) in progress with Mostafa Sabri

20 Exploration + V (x) in progress with Mostafa Sabri regular graph with (random) weights on the edges

21 Exploration + V (x) in progress with Mostafa Sabri regular graph with (random) weights on the edges some non-regular graphs? e.g. percolation graphs based on regular graphs

22 Wigner matrices Hemitian matrices of size N N, random iid entries. Law is centered, has a density and gaussian tails. Erdös, Schlein, Yau, Yin, Bourgade ( ), Tao-Vu : for any eigenvector φ, 1 φ N 1/2+ɛ φ 2 ( full delocalization, 2009) 2 η, ν > 0 s.t. B {1,..., N}, x B φ(x) 2 1 η = B νn. 3 QUE (2013) : for any fixed k, a N : {1,..., N} [ 1, 1], a N (x) φ (N) k (x) 2 a N δ a N N x with overwhelming probability.

23 Heavy-tailed matrices Symmetric matrices of size N N, random iid entries. Law is centered and has heavy tails : with 0 < α < 2. P( X > t) t α

24 Heavy-tailed matrices Symmetric matrices of size N N, random iid entries. Law is centered and has heavy tails : with 0 < α < 2. P( X > t) t α Ben Arous Guionnet have identified the limiting distribution of eigenvalues. Bordenave Guionnet have shown : If 1 < α < 2, for all but o(n) of the eigenvectors, there is delocalization in the sense that φ n ρ φ 2. If 0 < α < 2/3 and eigenvectors with eigenvalue λ large enough, there is localization : the mass of ( φ(x) 2 ) x=1,...,n is carried by at most n 1 δ entries.

25 Random non-symmetric matrices Vershynin-Rudelson 2014, Tao-Vu Matrices of size N N, random iid entries. Law is centered and has subexponential tail. Then with probability 1 n 1 t, all eigenvectors have 3/2 ln n9/2 φ Ct φ 2. n

26 Random band matrices Erdös-Knowles 2010 : symmetric matrices of size N N, band-width W N 6/7. Entries are iid, centered random variables, law has sub exponential tail. Then for most eigenvectors, the localization length is N.

27 Erdös-Rényi graphs Random graph with N vertices. Edges are chosen independently with probability p = p(n). The adjacency matrix is then a random N N symmetric matrix (containing only 0 and 1). Erdös, Knowles, Yau, Yin 2013 : if pn (ln N) C, then with high probability there is full delocalization of all eigenvectors : +QUE statement φ C (ln N)c N φ 2.

28 Random regular graphs with d N +. Dumitriu-Pal, Tran-Vu-Wang, Geisinger. Bauerschmidt-Knowles-Yau 2015 : if d N (log N) 4, then with proba 1 e ξ log ξ all eigenvectors have +QUE statement φ C ξ N φ 2.