The Matrix Dyson Equation in random matrix theory

Transcription

1 The Matrix Dyson Equation in random matrix theory László Erdős IST, Austria Mathematical Physics seminar University of Bristol, Feb 3, 207 Joint work with O. Ajanki, T. Krüger Partially supported by ERC Advanced Grant, RANMAT, No

2 INTRODUCTION Basic question [Wigner]: What can be said about the statistical properties of the eigenvalues of a large random matrix? Do some universal patterns emerge? H = h h 2... h N h 2 h h 2N... = (λ, λ 2,..., λ N ) eigenvalues? h N h N2... h NN N = size of the matrix, will go to infinity. Analogy: Central limit theorem: N (X + X X N ) N(0, σ 2 ) 2

3 Wigner Ensemble: i.i.d. entries H = (h jk ) real symmetric or complex hermitian N N matrix Entries are i.i.d. up to h jk = h kj (for j < k), with normalization Eh jk = 0, E h jk 2 = N. The eigenvalues λ λ 2... λ N are of order one: (on average) E N i λ 2 i = E N TrH2 = N ij E h ij 2 = If h ij is Gaussian, then GUE, GOE. 3

4 Global density: Semicircle Law Typical ev. gap N (bulk) Does semicircle law hold just above this scale? (= local semicircle law ) How do eigenvalues behave exactly on this scale? (= WDM universality) Global vs. local law Wigner s revolutionary observation: The global density 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 (GUE/GOE). 4

5 SINE KERNEL FOR CORRELATION FUNCTIONS Probability density of the eigenvalues: p(x, x 2,..., x N ) The k-point correlation function is given by p (k) N (x, x 2,..., x k ) := R N k p(x,... x k, x k+,..., x N )dx k+... dx N k = point correlation function: density ϱ Rescaled correlation functions at energy E (in the bulk, ϱ(e) > 0) p (k) E (x) := [ϱ(e)] kp(k) N ( E + x Nϱ(E), E + x 2 Nϱ(E),..., E + x k Nϱ(E) ) Rescales the gap λ i+ λ i to O(). 5

6 Local correlation statistics for GUE [Gaudin, Dyson, Mehta] lim N p(k) E (x) = det { S(x i x j ) } k i,j=, sin πx S(x) := πx Wigner-Dyson-Mehta universality: Local statistics is universal in the bulk spectrum for any Wigner matrix; only symmetry type matters. Solved for any symmetry class by the three step strategy [Bourgade, E, Schlein, Yau, Yin: ] Related results: [Johansson, 2000] Hermitian case with large Gaussian components [Tao-Vu, 2009] Hermitian case via moment matching. (Similar development for the edge, for β-log gases and for many related models, such as sample covariance matrices, sparse graphs, regular graphs etc). 6

7 Three-step strategy. Local density law down to scales /N (Needed in entry-wise form, i.e. control also matrix elements G ij the resolvent G(z) = (H z) and not only TrG) 2. Use local equilibration of Dyson Brownian motion to prove universality for matrices with a tiny Gaussian component 3. Use perturbation theory to remove the tiny Gaussian component. Steps 2 and 3 need Step as an input but are considered standard since very general theorems are available. Step is model dependent. 7

8 Models of increasing complexity Wigner matrix: i.i.d. entries, s ij := E h ij 2 are constant (= N ). (Density = semicircle; G diagonal, G xx G yy ) Generalized Wigner matrix: indep. entries, j s ij = for all i. (Density = semicircle; G diagonal, G xx G yy ) Wigner type matrix: indep. entries, s ij arbitrary (Density semicircle; G diagonal, G xx G yy ) Correlated Wigner matrix: correlated entries, s ij arbitrary (Density semicircle; G diagonal) 8

9 Variance profile and limiting density of states (DOS) j s ij = General variance profile s ij = E h ij 2 : not the semicircle any more. j s ij const = Density of states 9

10 Features of the DOS for Wigner-type matrices ) Support splits via cusps: (Matrices in the pictures represent the variance matrix) 2) Smoothing of the S-profile avoids splitting ( single interval) DOS of the same matrix as above but discontinuities in S are regularized 0

11 Universality of the DOS singularities for Wigner-type models Edge, E singularity Cusp, E /3 singularity Small-gap Smoothed cusp (2+τ)τ +(+τ+ (2+τ)τ) 2/3 +(+τ (2+τ)τ) 2/3 +τ 2 ( +τ 2 +τ) 2/3 +( +τ 2 τ) 2/3 τ := E gap, τ := E (minimum ) /3

12 Main theorems (informally) Theorem [Ajanki-E-Krüger, 204] Let H = H be a Wigner-type matrix with general variance profile c/n s ij C/N. Then optimal local law (including edge) and bulk universality hold. Theorem [Ajanki-E-Krüger, 206] Let H = H be correlated H = A + N W where A is deterministic, decaying away from the diagonal; W is random with EW = 0 and fast decaying correlation: Cov ( φ(w A ), ψ(w B ) ) C K φ ψ [ + dist(a, B)] K for all K and for any subsets A, B of the index set. Assume E u W v 2 c u 2 v 2 u, v. Then optimal local law and bulk universality hold. (Special translation invariant corr. structure: independently by [Che, 206] ) 2

13 Matrix Dyson Equation For any z C +, consider the equation (we set A = EH = 0) with the super-operator M = z + S[M], M = M(z) CN N () S[R] := E [ HRH ], S : C N N C N N Fact: [Girko, Pastur, Wegner, Helton-Far-Speicher] The MDE has a unique solution with Im M 0 and it is a Stieltjes transform of a matrix-valued measure M(z) = V (ω)dω π ω z, z C + Define the density of states ϱ(ω) := TrV (ω), πn ω R Theorem [AEK] (i) ϱ is compactly supported, Hölder continuous. (ii) V (ω) ϱ(ω) (iii) M xy has a fast offdiag decay away from the spectral edge. 3

14 Local law for the correlated case Theorem [AEK] In the bulk spectrum, ϱ(rz) c, we have G xy (z) M xy (z) with very high probability. N Im z, N TrG(z) N TrM(z) N Im z M is typically not diagonal, so G has nontriv off-diagonal component. We also have the usual Corollaries: Complete delocalization of corresponding eigenvectors Rigidity of bulk eigenvalues (ev s are almost in the /N-vicinity of the quantiles of the DOS). Wigner-Dyson-Mehta universality in the bulk 4

15 Derivation of the Matrix Dyson Equation G(z) := (H z) δ xy + zg xy = u h xu G uy Let U be a (large) neighborhood of {x, y}. Let H (U) be the removal of U rows/columns from H and G (U) is its resolvent. Using Thus G = G (U) G (U)[ H H (U)] G, G uy = v U δ xy + zg xy = u U w U G (U) uv h vwg wy, for u U. h xu G uy u,v U w U h xu G (U) uv h vwg wy Here G (U) uv is (almost) indep of h xu and h vw for w 2 U (for w U \ 2 U we use the decay of G wy) First sum is neglected, the uv sum in the second is close to its expectation. U U/2 w y x G (U) u v 5

16 The uv sum is close to its expectation u,v U h xu G (U) uv h vw u,v U Undoing the removal of U, we get i.e. δ xy + zg xy w E [ h xu h vw ] G (U) uv I + zg S[G]G ( ) S[G] G wy xw ( ) S[G (U) ] xw Thus G approximately solves the matrix Dyson equation (MDE) M = z + S[M], or I + zm = S[M]M. Key question: Stability of MDE under small perturbation. Then we could conclude that G M 6

17 Dyson equations and their stability operators Name Dyson Eqn For Stab. op Feature Wigner E h ij 2 = s ij = N Gen. Wigner j s ij = Wigner-type s ij arbitrary m = z + m m N TrG m 2 e e m = z + m m N TrG m 2 S m = z + Sm m x G xx m 2 S m = m sc is explicit Split S as S + e e m to be determined Corr. Wigner Eh xy h uw δ xw δ yu M = z + S[M] M xy G xy MS[ ]M Matrix eq. Super-op Gen. Wigner could be studied via a scalar equation only (in practice a vector eq. is also considered for G xx ) Wigner-type needs vector equation even for the density Corr. Wigner needs matrix equation. 7

18 Mechanism for stability I. Generalized Wigner For gen. Wigner, m is the Stieltjes tr. of the semicircle: m(z) cη, Im m(z) ϱ(e), z = E + iη The variance matrix S, with Se = e and a gap in Spec(S). m 2 S = e 2iϕ F, m = m e iϕ, F := m 2 S F is symmetric, Spec(F ) (, ) In the bulk ϕ Im m 0 i 2 Spec(e F) m 2 S = e 2iϕ F C ϕ - Spec (F) 0 2 Gap At the edge use the gap, the isolated eigenspace F e = m 2 e is treated separately. 8

19 Mechanism for stability II. Wigner-type m = z + Sm, S = s ij, m = (m i ) Why is ( m 2 S) invertible at all? [here (m 2 S) ij := m 2 i S ij] Take Im-part and symmetrize Im m m = η m + m S m Im m m Since Im m 0, by Perron-Frobenius, F := m S m cη Lemma. If F is self-adjoint with F f = F f and a gap, then U F Gap(F ) C F f, Uf Thus, we have stability (albeit weaker) m 2 S = e 2iϕ F, for any U unitary C (min ϕ j ) 2 9

20 Mechanism for stability III. Matrix Dyson Equation Theorem [AEK] Let S : C N N C N N be flat, i.e. and decay S[R] xy c N TrR S[R] C TrR, R 0 N C K R max ( + x y ) K, R max := max R ab ab For small D, a unique solution G = G(D) of the perturbed MDE = (z + S[M] ) M, = (z + S[G] ) G + D, that is linearly stable in strong sense G(D ) G(D 2 ) max C D D 2 max 20

21 Matrix stability operator Define the sandwiching operator on matrices: C R [T ] := RT R Lemma: M = M(z) be the solution to MDE, then MS[ ]M = C M S with C depending on M in a controlled way. C [ϱ(z) + dist(z, supp(ϱ)] 00 Key: find the right symmetrization F despite the noncommutative matrix structure. Need the analogue of m = e iϕ m, F = m S m, m 2 S = e 2iϕ F 2

22 Answer: Polar decompose M into a commuting quarter magnitude W > 0 and a phase U (unitary) M = C ImM C W [U ] = Im M W U W Im M W := + ( ImM Re M ImM ) 2 4, U := Re M i ImM ImM W 2 Define F := C W C ImM SC ImM C W Then F is selfadjoint (wrt. HS scalar product), has a unique normalized eigenmatrix F with e.v. F and a spectral gap: C M S U F Gap(F) F F, UF U Then we prove F F, UF U c, Gap(F) c with some c = c(ϱ). 22

23 Summary We gave a quantitative analysis of the solution of the Matrix Dyson Equation and its stability. For correlated random matrices with short range correlation in both symmetry classes we proved Optimal local law in the bulk Wigner-Dyson-Mehta bulk universality 23

24 Outlook Add arbitrary external field (A = EH) work in progress Cusp analysis for Wigner type work in progress Edge analysis for MDE work in progress No. of intervals in suppϱ in terms of block structure of S or S? 24