Isotropic local laws for random matrices

Transcription

1 Isotropic local laws for random matrices Antti Knowles University of Geneva With Y. He and R. Rosenthal

2 Random matrices Let H C N N be a large Hermitian random matrix, normalized so that H. Some motivations: Quantum mechanics: Hamilton operator of a disordered quantum system (heavy nuclei, itinerant electrons in metals, quantum dots,...). Multivariate statistics: sample covariance matrix. Goal: Analysis of eigenvalues λ λ 2... λ N and eigenvectors of H. u, u 2,..., u N S N

3 The key questions () Global eigenvalue distribution. Asymptotic behaviour of the empirical distribution N N i= δ λ i. λ i+ λ i (2) Local eigenvalue distribution. Asymptotic behaviour of individual eigenvalues. λ Examples: distribution of the gaps λ i λ i+ or the largest eigenvalue λ. (3) Distribution of eigenvectors. Localization / delocalization of the eigenvectors. Distribution of the components v, u i.

4 Global and local laws The Green function G(z).= (H zi) is the right tool to address the questions () (3). Writing z = E + iη C +, we have Im Tr G(z) = N N N i= η (λ i E) 2 + η 2 Observation: η = Im z is the spectral resolution. Global law: control of G(z) for η. Local law: control of G(z) for η N. λ i η E To answer the questions (2) and (3), one needs a local law.

5 Deterministic equivalent of Green function One usually needs control of G(z) as a matrix, and not just of N Tr G(z). Goal: There is a deterministic matrix M(z), the deterministic equivalent of G(z), such that G(z) M(z) is small for η N with high probability. What does G M small mean? Canonical notions of smallness (operator topologies): control of (i) v, (G M)w, (ii) (G M)v, (iii) G M for all deterministic v, w S N. In fact, already for H = GUE it is easy to see that (ii) and (iii) blow up. Control of (i) is the strongest one can hope for (isotropic control of G).

6 Example: Wigner matrices The entries (H ij. i j N) are independent and satisfy The deterministic equivalent is where EH ij = 0, E H ij 2 = N. m(z).= M(z) = m(z)i 2 2 dx 4 x 2 2π is the Stieltjes transform of the semicircle law. x z

7 Some history Local law η N for Wigner matrices: (a) Tr(G M) [Erdős, Schlein, Yau; 2009] (b) (G M) ij [Erdős, Yau, Yin; 200] (c) v, (G M)w [Knowles, Yin; 20] More general models (sparse random matrices, covariance matrices, deformed matrices,...): [Ajanki, Erdős, Knowles, Krüger, Lee, Schnelli, Yau, Yin,... ] Two key steps in all proofs: Deterministic step: stability of self-consistent equation. Identify M as the solution of a self-consistent equation Π(M) = 0. Prove that Π(Q) 0 = Q M. Stochastic step: derivation of the self-consistent equation. Prove that Π(G) 0 with high probability.

8 Derivation of the self-consistent equation: folklore Use Schur s complement formula to write G ii = z H ii k,l i H ikg (i) kl H li and large deviation estimates to show, with high probability, H ii 0 and k,l i Average over i. H ik G (i) kl H li H ik G (i) kl H ki N k i k i, G (i) kk N G kk. Works very well for Wigner matrices and some generalizations theoreof. Problems: (i) Matrix entries have to be independent. (ii) Expectation of H has to be diagonal. (iii) Does not give control of v, (G M)w. This requires an additional, difficult, step. k

9 Alternative approach [He, K, Rosenthal; 206] New way to derive self-consistent equations, overcoming all of the above problems: (i) Admits a very general relationship between matrix entries and the independent random variables. (Can also handle models with short-range correlations, [Erdős, Krüger, Schröder; 207].) (ii) Completely insensitive to the expectation of H. (iii) Yields control of v, (G M)w from the outset. Key idea: instead of working on entire rows and columns (Schur s formula), work on individual entries (resolvent/cumulant expansion).

10 Resolvent / cumulant expansion Resolvent expansion in individual entries: (H (ij) ) kl.= {i,j} ={k,l} H kl, G (ij) (z).= (H (ij) zi). Starting point: trivial identity I + zg = HG. Then write E(HG) ii = E j = E j H ij G ji H ij ( G (ij) ji G (ij) jj H jig (ij) ii ) G (ij) ji H ij G (ij) ji + = E j = E j N G(ij) jj G(ij) ii + N G jjg ii +. Note: resolvent expansion is used twice: G G (ij) G.

11 The resulting algebra is beautifully summarized by the cumulant expansion E[h f(h)] = l k=0 k! C k+(h) E[f (k) (h)] + R l, C k (h).= k t t=0 log E[e th ]. [Khorunzhy, Khoruzhenko, Pastur; 996] Performs essentially the same as the resolvent expansion but more tidily. In applications, h = H ij and f(h) is a polynomial of resolvents. For example, the previous resolvent calculation is replaced by E(HG) ii = j = j l E[H ij G ji ] = j k=0 [ ] N E G ji + = H ij j [ ( ) k k! C k+(h ij )E G ji] + H ij N E[ G jjg ii G ji G ij ] + Second term small by Cauchy-Schwarz and Ward identity j G ij 2 = η Im G ii.

12 Sketch of results Illustrative model: general mean-field model with independent entries. The entries (H ij. i j N) are independent and satisfy Var(H ij ) = O( N ). Split H = W + A where A.= EH, and define the map Π z (M).= I + zm + S(M)M AM, S(M).= E[W MW ]. Then for z C + the equation Π z ( ) = 0 has a unique solution M(z) with positive imaginary part the deterministic equivalent of G for this model. We prove that for all η N v, Π(G)w (Optimal in bulk and at edges.) Im M + Nη Nη. This deals with the stochastic step derivation of self-consistent equation. Conclude proof of local law by the deterministic step stability analysis of self-consistent equation [Lee, Schnelli; 203], [Ajanki, Erdős, Krüger; 206].

13 How to start the proof Let P vw.= v, Π(G)w, where Π(G) = I + zm + S(M)M AM = W G + S(G)G. By Markov s inequality, it suffices to estimate E P vw 2p = E [ (W G) vw Pvw p P p ] [ vw + E (S(G)G)vw Pvw p P p vw]. Apply the cumulant expansion to the first term by writing (W G) vw = i,j v i W ij G jw. The leading term from k =, [( ) ] E v i G jj G iw Pvw p P p vw, i,j cancels the term E [ (S(G)G) vw P p vw P p vw]. Everything else has to be estimated main work!