Spectral Universality of Random Matrices

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1 Spectral Universality of Random Matrices László Erdős IST Austria (supported by ERC Advanced Grant) Szilárd Leó Colloquium Technical University of Budapest February 27, 2018 László Erdős Spectral Universality of Random Matrices 1

2 The question that mathematicians failed to ask What can be said about the statistical properties of the eigenvalues of a large random matrix? Do some universal patterns emerge? Eugene Wigner (1954) h 11 h h 1N h 21 h h 2N H =... = (λ 1, λ 2,..., λ N ) eigenvalues? h N1 h N2... h NN Why is it relevant for physics? László Erdős Spectral Universality of Random Matrices 2

3 Quantum systems Quantum mechanical system is described by a state space l 2 (S) (e.g. S = {, } for a spin or S = Z 3 for an electron in a metallic lattice); an S S symmetric (selfadjoint) matrix (operator) H, the Hamilton operator; Matrix elements H s,s describe quantum transition rates from s to s. Eigenvalues of H (real) are the possible energy levels of the system Time evolution is given by ψ t = e ith ψ 0 Disordered: if H s,s is random = Random matrix. László Erdős Spectral Universality of Random Matrices 3

4 Universality conjecture for disordered quantum systems: A disordered quantum system with sufficient complexity exhibits one of the following two behaviors: Insulator (typically at strong disorder) Localized eigenvectors lack of transport nearby eigenvalues are independent (Poisson local spectral statistics). Conductor (typically at weak disorder) Delocalized eigenvectors transport (via quantum diffusion) eigenvalues are strongly correlated (Wigner-Dyson local statistics) At first sight, localization is surprising (Anderson). Still, mathematically it is more accessible (Fröhlich-Spencer, Aizenman-Molchanov, Minami,...). The perturbative conductor regime is very hard. Anderson metal-insulator quantum phase transition László Erdős Spectral Universality of Random Matrices 4

5 Two popular models to study the dichotomy (1) Random Schrödinger operators: in lattice box S := [1, L] d Z d In d = 1 it corresponds to a narrow band matrix with i.i.d. diagonal: v v 2 1 H = +. v x = 1.. x v L v L Follows insulator behavior László Erdős Spectral Universality of Random Matrices 5

6 (2) Mean field Wigner random matrices: On the set S = {1, 2,..., N} H = (H xy ), H = H EH xy = 0. entries are identically distributed and independent up to symmetry. h 11 h h 1N h 21 h h 2N H =... h N1 h N2... h NN H models a mean-field hopping mechanism with random quantum transition rates. No spatial structure. Follows conductor behavior Key physics conjecture: Interpolation between random Schrödinger and Wigner gives rise Anderson phase transition in band matrix ensembles László Erdős Spectral Universality of Random Matrices 6

7 Band matrices: interpolation between Anderson and Wigner S = [1, L] d Z d lattice box Entries of H = H are indep but not identically distr. E H xy 2 = 1 ( x y ) W d f W W is the bandwidth (range of interaction) H = Band matrix in d = 1 physical dim. W = O(1) [ Anderson] W = L [Wigner] Facts : Transition occurs at W L 1/2 (d = 1) SUSY [Fyodorov-Mirlin, 1991] W log L (d = 2) RG scaling [Abrahams, 1979] W W 0 (d) (d 3) extended states conjecture Math. results in d = 1: localized if W L 1/8 [Schenker] deloc. if W L 4/5 [E-Knowles-Yau] László Erdős Spectral Universality of Random Matrices 7

8 Wigner matrix ensemble H = (h jk ) 1 j,k N complex hermitian or real symmetric N N matrix h jk = h kj (for j < k) are indep. random variables with normalization Eh jk = 0, E h jk 2 = 1 N. The eigenvalues λ 1 λ 2... λ N are of order one: (on average) E 1 λ 2 i = E 1 N N Tr H2 = 1 E h ij 2 = 1 N i ij László Erdős Spectral Universality of Random Matrices 8

9 Wigner semicircle law Let H be a Wigner matrix with eigenvalues λ 1, λ 2,..., λ N. Density of evalues/states (DOS) on the global (macroscopic) scale: 1 N #{λ i [a, b]} b a ϱ(x)dx, ϱ(x) = 1 (4 x2 ) + 2π holds for any fixed [a, b] interval. What about shrinking intervals as 1 N b a 1? Local Law! László Erdős Spectral Universality of Random Matrices 9

10 Rigidity of eigenvalues What are the typical fluctuations of the eigenvalues? Your probabilist s/physicist s instinct would predict λ i γ i 1 N where γ i s are the corresponding i-th N-quantiles of ϱ: γi ϱ(x)dx = i N which were correct by CLT if λ i s were independent. FACT: Eigenvalues are strongly correlated and are very rigid: log N λ i γ i N László Erdős Spectral Universality of Random Matrices 10

11 Global vs. local statistics Typical eigenvalue spacing is of order 1 N in the bulk spectrum How do eigenvalues behave on this scale? Gap statistics: Nϱ(λ i )(λ i+1 λ i ) is normalized to have expectation 1. Wigner s revolutionary observation: DOS may be model dependent, but the gap statistics is very robust, it depends only on the symmetry class (hermitian or symmetric). In particular, it can be determined from the Gaussian case which has been computed in the 60 s (Dyson s sine kernel) Formulated more precisely as the Wigner-Dyson-Mehta conjecture in 60 s László Erdős Spectral Universality of Random Matrices 11

12 Wigner surmise and level repulsion for GUE ( ) P Nϱ(λ i )(λ i+1 λ i ) = s + ds 32 π 2 s2 e 4s 2 /π ds, for λ i in the bulk (repulsive correlation!) Histogram of the rescaled gaps for GUE matrix with N = 3000 László Erdős Spectral Universality of Random Matrices 12

13 Resolution of the Wigner-Dyson-Mehta universality conjecture Theorem. The gap distribution of a Wigner matrix in the bulk spectrum is universal, it depends only on the symmetry type and is independent of the distribution of the matrix elements. In particular, it coincides with that of the computable Gaussian case. [Johansson, 2000] Hermitian case with large Gaussian component [E-Peche-Ramirez-Schlein-Yau, 2009] Hermitian case with smoothness [Tao-Vu, 2009] Hermitian case via moment matching [E-Schlein-Yau, 2009] Symmetric and hermitian cases via Dyson Brownian motion [E-Schlein-Yin-Yau, 2010] Generalized Wigner matrices [E-Knowles-Yin-Yau, 2012] Sparse matrices, Erdős-Rényi graphs. [E-Yau, 2012] Single gap universality [Bourgade-E-Yin-Yau, 2014] Fixed energy universality [Ajanki-E-Krüger, 2015] General variance profile beyond semicircle [Ajanki-E-Krüger, 2016] Beyond independence: general correlations included. [E-Krüger-Schröder, 2017] Long range correlation László Erdős Spectral Universality of Random Matrices 13

14 Basic approach: three step strategy [E-Schlein-Yau 2009] 1. Prove rigidity just above the optimal scale: λ i γ i N ε N (in the bulk) 2. Use the fast local equilibration property of the Dyson Brownian motion to prove universality for matrices with a tiny Gaussian component 3. Use perturbation theory to remove the Gaussian component Step 2 and Step 3 need Step 1 as an input, but very general theorems are available. Most of our activity lately aimed at Step 1 for models of increasing complexity to go beyond Wigner matrices (e.g. DOS is not the semicircle). Start with an example. László Erdős Spectral Universality of Random Matrices 14

15 General variance profile instead of semicircle s ij = 1-2 j General variance profile s ij = E h ij 2 : not the semicircle any more. j s ij const = Density of states László Erdős Spectral Universality of Random Matrices 15

16 Features of the DOS for Wigner-type matrices 1) Support splits via cusps: (Matrices in the pictures represent the variance matrix) 2) Only square root singularities at the edges and cubic root singularities at the cusps appear. No other singularity occurs! László Erdős Spectral Universality of Random Matrices 16

17 (Mean field) models of increasing complexity Wigner matrix: i.i.d. entries, s ij := E h ij 2 are constant (= 1 N ). Density is the semicircle. [E-Schlein-Yau-Yin, ], [Tao-Vu, 2009] Wigner type matrix: indep. entries, s ij arbitrary (no semicircle) [Ajanki-E-Krüger, 2015] Correlated Wigner matrix: correlated entries, s ij arbitrary [Ajanki-E-Krüger, 2015] [Che, 2016], [Ajanki-E-Krüger, 2016], [E-Krüger-Schröder, 2017] Oskari Ajanki Torben Krüger (Bonn) Dominik Schröder László Erdős Spectral Universality of Random Matrices 17

18 Matrix Dyson Equation How to compute the density in general? Solve the Dyson equation! Fact 1: Given a probability measure ϱ on R, its Stieltjes transform is ϱ(x)dx m(z) = x z, z C +, Inverse ϱ(x) = 1 Im m(x + i0) π R Fact 2: Solve the Matrix Dyson Equation for M C N N I = Mz + S[M]M, with Im M 0 (MDE). Here S[R] := EHRH is the self-energy operator. Then M(z) 1 1 and Tr M(z) = H z N R ϱ(dx) x z Fact 3: Trace of the resolvent is the St. tr. of the eigenvalue density: 1 N Tr M(z) 1 N Tr 1 H z = N λ i i z = N i δ(x λ i) R x z Its inverse St. transform gives the density ϱ. Main technical work: study the properties of the innocent looking MDE. László Erdős Spectral Universality of Random Matrices 18

19 Derivation of MDE G(z) := (H z) 1 I + zg = HG Write it as I + (z + S[G])G = D with D := HG + S[G]G Note that MDE is equivalent to the same eq. with D = 0 I + (z + S[M])M = 0 (MDE) To conclude G M, need to show that (i) D is small, i.e. G approx. satisfies MDE. (ii) The MDE is stable. In the Gaussian case, a simple integration by parts suffices: ED = E [ HG + S[G]G ] [ = E Ẽ[ HG H]G ] + S[G]G = 0 and higher moments E D p are small by a similar argument. In the general case, one can use cumulant expansion László Erdős Spectral Universality of Random Matrices 19

20 Main result on correlated Wigner matrices Thm. [Ajanki-E-Krüger 16, E-Krüger-Schröder 17] Consider a general hermitian random matrix with a decaying correlation structure: H = A + 1 N W (A is deterministic, W is random) such that EW = 0, A C Cov ( φ(w A ), ψ(w B ) ) C[1 + dist(a, B)] 12 for any A, B [N] 2 κ(φ 1 (W A1 ), φ 2 (W A2 ),...) {A j,a k } T min Cov(φ j, φ k ) E u, W v 2 c u 2 v 2 u, v. Then optimal local law and its usual corollaries hold in the bulk. Next pages: more precisely László Erdős Spectral Universality of Random Matrices 20

21 Distance for index sets in the condition on the correlation decay Cov ( φ(w A ), ψ(w B ) ) C K φ ψ [1 + dist(a, B)] 12 for any A, B S S, assuming the usual metric on the set S = {1, 2,..., N} of indices. Here W A = {W ij : (i, j) A}. ( d(a, B) ) A A B B László Erdős Spectral Universality of Random Matrices 21

22 Clustering of higher order cumulants The higher order cumulants of (functions of) matrix elements supported in disjoint sets A 1, A 2,... have a decay given by the covariance decay along a minimal spanning tree κ(φ 1 (W A1 ), φ 2 (W A2 ),...) {A j,a k } T min Cov(φ j, φ k ) Standard property in statistical physics (high temperature regime). László Erdős Spectral Universality of Random Matrices 22

23 Optimal local law: In the bulk spectrum, ϱ(rz) c, we have v, [G M]u v u N Im z, 1 N Tr B(G M) B N Im z with very high probability, where M solves the Dyson equation. M is typically not diagonal, so G has nontriv off-diagonal component. Corollaries (i) Eigenvector delocalization: v N 1/2 v 2 for Hv = λv (ii) Eigenvalue rigidity: λ i γ i N 1 in bulk (iii) Universality of local correlation functions and gaps. (Special translation invariant corr. structure: independently by [Che, 2016] ) László Erdős Spectral Universality of Random Matrices 23

24 Bulk universality: Dyson sine-kernel statistics holds on the level of individual eigenvalues: László Erdős Spectral Universality of Random Matrices 24

25 Structured models with independent entries General sample covariance (or Gram) matrix: H = XX t, xij indep. [Alt-E-Kru ger, 2016], [Alt 2017] Local inhomog. circular law: H has indep. entries. No hermitian symmetry, spectrum is in a disk on the complex plane. [Alt-E-Kru ger, 2016] Block (Kronecker) matrices: H = P i αi Xi, Xi have indep. entries [Alt-E-Kru ger-nemish 2017] Applications: decay of critically tuned random ODE s from neuroscience [E-Kru ger-renfrew, 2017] Johannes Alt La szlo Erdo s Torben Kru ger (Bonn) Yuriy Nemish David Renfrew Spectral Universality of Random Matrices 25

26 ODE s with random coefficients Consider the linear differential equation d u = u + gxu, dt u = u(t) CN with (random) coefficient matrix X C N N with independent entries and coupling constant g > 0. Solution (via matrix exponential) is u(t) = e t( 1+gX) u 0 For small g: exponential decay; for large g: exponential blow up. For a critical value of g the system has universal power law decay, depending on the symmetry of X: { t 1/2 if X has indep. entries, no symmetry u(t) 2 t 3/2 if X has indep. entries with hermitian symmetry Predicted by Chalker and Mehlig (PRL 2000) for the Gaussian case by non-rigorous diagrammatic expansion. László Erdős Spectral Universality of Random Matrices 26

27 Random matrices via random unitaries Suppose A and B are (fixed) Hermitian matrices (two Hamiltonians). What is the spectrum of A plus B when added up in random relative bases? Free convolution! Addition of random matrices: H = A + U BU, U is Haar unitary [Voiculescu 1992, Bao-E-Schnelli, 2015, 2016, 2017] Local single ring theorem: H = U ΣV ; where Σ is fixed, U, V are indep. Haar unitaries [Guionnet-Krishnapur-Zeitouni 2011, Bao-E-Schnelli 2016] Zhigang Bao (HKUST) La szlo Erdo s Kevin Schnelli (KTH) Spectral Universality of Random Matrices 27

28 Free convolution via examples Compare with the usual convolution from (commutative) probability theory for the distribution of the sum of two indep. random variables. semicircle semicircle = semicircle Bernoulli /2 1/2 = László Erdős Spectral Universality of Random Matrices 28

29 Bernoulli Bernoulli 2 1/2 1/2 1/2 1/2 = three point masses three point masses 3 1/4 1/2 1/4 1/4 1/2 1/4 = László Erdős Spectral Universality of Random Matrices 29

30 Free additive convolution Let A, B be large hermitian matrices with spectral densities µ A (dx) = 1 δ(x a i ), µ B (dx) = 1 δ(x b i ), N N i i and denote their Stieltjes transforms by m A (z) = 1 N i 1 a i z, m B(z) = 1 N i 1 b i z. Then the St. transform of A + UBU is (approx) given by the free additive convolution of µ A B whose St. transform is the solution to the subordination equations: [Voiculescu 93, Biane 98] m A B (z) := m A (ω B (z)) = m B (ω A (z)), [m A (ω B (z))] 1 = ω A (z) + ω B (z) z. where ω A, ω B are analytic functions with positive imaginary parts. Theorem [Bao-E-Schnelli] Local version of Voiculescu s theorem holds. László Erdős Spectral Universality of Random Matrices 30

31 Dream 1: Band matrices Recall the universality conjecture for disordered systems So far spectral universality has been proven only for mean field models that are naturally in the delocalized regime. We are very far from covering any nontrivial delocalized systems. One prominent (simplest) example is the 1d band matrix with W N. Only two results exist: The 2-point correlation function of det(h E) converges to the sine kernel [T. Shcherbina 2014] with rigorous SUSY Wigner-Dyson statistics hold for W cl (macroscopic bandwidth) [Bourgade-E-Yau-Yin, 2016] László Erdős Spectral Universality of Random Matrices 31

32 Dream 2: Anderson transition Anderson model (of electron propagation in disordered lattice) H = + λ x Z d ω x δ x, {ω x } x Z d i.i.d., E ω 2 = 1 Extended states conjecture: For d 3 and 0 < λ λ c (d), the spectrum of H is a.c. (in the bulk) the corresponding generalized e.functions are delocalized (not in l 2 ) the gap statistics follow GOE. For the moment, even the existence of such regime is too hard. László Erdős Spectral Universality of Random Matrices 32

33 Dream 3: heavy nuclei Wigner s original motivation was to predict energy level statistics of heavy nuclei and show that they are universal To verify this rigorously would be very hard, since there is no randomness apart from sampling the energy gaps. László Erdős Spectral Universality of Random Matrices 33

34 Dream 4: Riemann zeros The rescaled zeros of the Riemann zeta function on the critical line follow RMT statistics. (Proven in a restricted class of test functions in 1972 by Montgomery) László Erdős Spectral Universality of Random Matrices 34

35 Thanks for the attention! László Erdős Spectral Universality of Random Matrices 35