The Matrix Dyson Equation in Random Matrix Theory

Home , Freeman Dyson

The Matrix Dyson Equation in random matrix theory

L´aszl´oErd˝os IST, Austria

Vienna Doctoral School

3rd Summer School in Math Physics

Weissensee, Austria, Sep 16–22, 2018

Partially supported by ERC Advanced Grant, RANMAT, No. 338804 1 Institute of Science and Technology (IST) Austria

• Newly founded (2007) research institute near Vienna, Austria

• Mathematics, CS, Physics, Biology + interdisciplinary research

• 50 research groups so far, 8 in mathematics

• Only graduate school, no undergraduates

• Fully English speaking

• 22 min from Heiligenstadt and Viennese students are welcome!!!

• For more, see www.ist.ac.at (Seminars, job opportunities etc.)

2 IST AUSTRIA MAIN CAMPUS NEAR VIENNA

3 IST AUSTRIA CENTRAL BUILDING

4 THEORY BUILDING (MATH/CS)

5 DISCUSSION AREA IN THE MATH FLOOR

6 “Perhaps I am now too courageous when I try to guess the distribution of the distances between successive levels (of en- ergies of heavy nuclei). Theoretically, the situation is quite simple if one attacks the problem in a simpleminded fash- ion.The question is simply what are the distances of the characteristic values of a symmetric matrix with random co- eﬃcients.”

Eugene Wigner, 1956

Nobel prize 1963

7 Plan of the lectures

1. Introduction, Wigner’s vision, concepts, correlation functions, scales, WDM universality, Wigner vs. invariant ensembles.

2. Three step strategy, Dyson Brownian motion, Local law, Stieltjes transform, Helﬀer-Sj¨ostrandformula,

3. Models of increasing complexity, singularity structure of the density, motivations from physics: universal dichotomy in disordered quantum systems, band matrices.

4. Some ideas of proofs: moment vs. resolvent method. Schur formula, derivation of the Dyson equation. Large deviation es- timates, Ward identity. Analysis of the Dyson equation, exis- tence/uniqueness, self-energy operator, Perron-Frobenius theorem, 1/3-H¨olderregularity. Matrix Dyson equation, symmetric polar decomposition.

8 LECTURE 1: 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?

  h11 h12 . . . h1N    h21 h22 . . . h2N  H =  . . .  =⇒ (λ1, λ2, . . . , λN ) eigenvalues?  . . .  hN1 hN2 . . . hNN

N = size of the matrix, will go to inﬁnity.

1 2 Analogy: Central limit theorem: √ (X1 + X2 + ... + X ) ∼ N(0, σ ) N N

9 Wigner Ensemble: i.i.d. entries

H = (hjk) real symmetric or complex hermitian N × N matrix

¯ Entries are i.i.d. up to hjk = hkj (for j < k), with normalization

2 1 Ehjk = 0, E|hjk| = . N

The eigenvalues λ1 ≤ λ2 ≤ ... ≤ λN are of order one: (on average)

1 X 2 1 2 1 X 2 E λi = E TrH = E|hij| = 1 N i N N ij

If hij is Gaussian, then GUE, GOE.

10 Global vs. local law

Global density: Semicircle Law 1 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)

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).

11 Wigner surmise and level repulsion

π 2 N%(λ )(λ − λ ) = s + ds ' s e−πs /4 ds PGOE i i+1 i 2 32 2 −4s2/π PGUE N%(λi)(λi+1 − λi) = s + ds ' s e ds , π2 for λi in the bulk (repulsive correlation depending on the symmetry!)

1.0

0.30

0.8 0.25

0.20 0.6

0.15 0.4

0.10

0.2

0.05

0.0 0.00 -2 -1 0 1 2 0.5 1.0 1.5 2.0 2.5

Side note: Wigner surmise guessed on a 2 × 2 matrix calculation and is only an approximation to the true gap distribution given by a Fredholm determinant of the Dyson’s sine-kernel [Gaudin, Mehta, Dyson]

12 E. Wigner (1955): The excitation spectra of heavy nuclei have the same spacing distribution as the eigenvalues of GOE. Experimental data for excitation spectra of heavy nuclei: (238U)

Typical Poisson statistics:

Typical GOE random matrix eigenvalues

13 Level spacing (gap) histogram for diﬀerent point processes.

NDE – Nuclear Data Ensemble, resonance levels of 30 sequences of 27 diﬀerent nuclei. 14 Zeros of the Riemann-zeta function (detour)

1 1 δn = γ +1 − γn, γn = γn log γn, ζ( + iγn) = 0 bn b b 2π 2

15 16 Pair correlation functions (sine kernel)

17 Sine kernel for correlation functions

Probability density of the eigenvalues: p(x1, x2, . . . , xN )

The k-point correlation function is given by (k) Z pN (x1, x2, . . . , xk) := p(x1, . . . xk, xk+1, . . . , xN )dxk+1 ... dxN RN−k k = 1 point correlation function: density %

Rescaled correlation functions at energy E (in the bulk, %(E) > 0)

( ) 1 ( ) x x x p k (x) := p k E + 1 ,E + 2 ,...,E + k E [%(E)]k N N%(E) N%(E) N%(E)

Rescales the gap λi+1 − λi to O(1).

18 Local correlation statistics for GUE [Gaudin, Dyson, Mehta]

(k) n ok sin πx lim p (x) = det S(xi − xj) ,S (x) := N→∞ E i,j=1 πx

Special k = 2 case: Pair correlations.

1 (2) x x lim p E + 1 ,E + 2 N→∞ [ρ(E)]2 N Nρ(E) Nρ(E)

n o2 sin πx = det S(x − x ) ,S(x) := , |E| < 2 i j i,j=1 πx

sin π(x − x )2 = 1 − 1 2 =⇒ Level repulsion π(x1 − x2)

19 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: 2009-2014]

Related results: [Johansson, 2000] Hermitian case with large Gaussian components [Tao-Vu, 2009] Needs four 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).

20 ********************************************************

Summary of the ﬁrst lecture

— We study eigenvalue statistics of large N × N random matrices.

— Wigner matrices: i.i.d. entries (up to symmetry)

— Normalization: λj ∼ O(1), gap = λi+1 − λi ∼ 1/N (bulk)

— Scales: Global [O(1)], Mesoscopic, Microscopic O(1/N).

— Wigner’s fundamental vision: Global density may be model dependent, but microscopic statistics is universal, depends only on the symmetry type [Wigner-Dyson-Mehta universality]

21 — Semicircle law holds only for Wigner matrices. Sample covariance matrices have diﬀerent global density (Marchenko–Pastur law).

— Local statistics (sine kernel, determinant structure of higher order correlation functions) were found in the 60’s for GOE/GUE.

— WDM universality has recently been proven for Wigner matrices via the three-step strategy.

— Main input for the three step strategy is the optimal local law (LLN on optimal mesoscopic scales)

********************************************************

22 Lecture 2: MOTIVATIONS

• Physics: nuclear physics (Wigner), disordered quantum systems

• Statistics: Wishart ensemble, sample covariance matrices (XXt), principal component analysis, Tracy Widom law

• Wireless communication; channel capacity [Tulino-Verdu], [Hachem- Loubaton-Najim], [Alt-E-Kr¨uger]

• Neural networks, ODE’s with random coeﬃcients. [Chalker-Mehlig], [E-Kruger-Renfrew]

• Quantum chaos, Quantum unique ergodicity [Bourgade-Yau]

23 Quantum systems

Quantum mechanical system is described by

2 3 • a state space ` (Σ) (e.g. Σ = {↑, ↓} for a spin, or Σ = Z for an electron in a metallic lattice);

• an Σ×Σ symmetric (selfadjoint) matrix (operator) H, the Hamil- ton operator;

• Matrix elements Hx,x0 describe quantum transition rates from x to x0.

• Eigenvalues of H (real) are the energy levels of the system

itH • Time evolution is given by ψt = e ψ0

Disordered: if Hx,x0 is random =⇒ Random matrix. 24 Another class: Invariant ensembles (detour)

Unitary ensemble: Hermitian matrices with density

P(H)dH ∼ e−Tr V (H)dH Invariant under H → UHU−1 for any unitary U (GUE) Joint density function of the eigenvalues is explicitly known P Y β − j V (λj) p(λ1, . . . , λN ) = const. (λi − λj) e i

Correlation functions can be explicit computed via orthogonal polynomials due to the Vandermonde determinant structure. large N asymptotic of orthogonal polynomials =⇒ local eigenvalue statistics indep of V . But density of e.v. depends on V . 25 Many previous results for classical invariant ensembles

Dyson (1962-76), Mehta (1960-1990) classical Gaussian ensembles via Hermite polynomials

General case by Deift etc. (1999), Pastur-Schcherbina (2008), Bleher-Its (1999), Deift etc (2000-, GOE and GSE), Lubinsky (2008)

All these results are limited to invariant ensembles. If the condition on Gaussian distribution is dropped (Wigner ensembles), the ensemble is not invariant.

The only result for the non-invariant case is by Johansson 2001, (Ben Arous-Peche, 2005) where sine kernel is proven for the Hermitian Wigner ensembles with a substantial Gaussian component: √ √ H = 1 − tH0 + tV, t > 0,H0 is Wigner V is GUE Method: saddle point analysis on an explicit formula (Brezin-Hikami) valid only for Hermitian matrices. 26 Three-step strategy

1. Local density law down to scales 1/N

(Needed in entry-wise form, i.e. control also matrix elements Gij the resolvent G(z) = (H − z)−1 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 1 as an input but are considered standard since very general theorems are available. [E-Schlein-Yin-Yau], .... most recent: [Landon-Yau]

Step 1 is model dependent. 27 Formulation of the local law via resolvents

−1 Resolvent: G(z) = (H − z) , at spectral parameter z ∈ C+.

1 Claim: Resolvent is asymptotically deterministic for η := Im z N .

Theorem: There exists a deterministic matrix Mxy(z) such that ε N Gxy(z) − Mxy(z) ≤ √ (Entrywise law) Nη ε 1 1 N TrBG(z) − TrBM(z) ≤ kBk (Average law) N N Nη Warning: Resolvent becomes deterministic! 1 ≈ M, but H 6≈ z + M−1 H − z since the RHS is only ﬂuctuation!

Reason: the entrywise closeness is not a suﬃciently strong topology to make inverse continuous. 28 Why resolvent?

By spectral theorem (of hermitian matrices/self-adjoint operators)

1 N |v ihv | N 1 G = = X i i , TrG = X H − z i=1 λi − z i=1 λi − z where Hvi = λivi are the evalues/evectors of H.

Note that 1 1 η 1 Im Tr ( + ) = X = X ( ) G E iη 2 2 δη λi − E N N i |λi − E| + η N i so eigenvalue density on scale η is identiﬁed via the average law with B = 1: 1 1 Im TrG(E + iη) ≈ Im TrM(E + iη) N N

Behind this formalism: Stieltjes transform.

29 What is M? Matrix Dyson equation

Full generality: Given H = H∗ random matrix, let

h i N×N N×N A := EH, S[R] := E (H − A)R(H − A) , S : C → C be the expectation matrix and the variance operator.

Note that S is symmetric wrt. the natural Hilbert-Schmidt scalar product and positivity preserving.

For any z ∈ C+, consider the equation

1 N×N − = z − A + S[M],M = M(z) ∈ C (1) M

FACT: This equation has a unique solution satisfying Im M > 0.

This M will approximate G. 30 Dyson equation for Wigner matrices

2 1 Wigner matrix: independent entries with Ehij = 0 and E|hij| = N .

Thus A = EH = 0 and we compute the matrix elements of S[R] h i X 1 1 S[R]ab = E (HRH)ab = E haiRijhjb = δab TrR + (1 − δab) Rba ij N N Clearly 1 − = z − A + S[M] M has a diagonal solution, M(z) = m(z)Id where m(z) solves 1 − = z + m, Im m > 0 m

Homework: m(z) = Stieltjes transform of Wigner’s semicircle law

Z ρ(x)dx 1 q 2 m(z) = , ρ(x) = (4 − x )+ R x − z 2π Stieltjes transform

Def: Let µ be a probability measure on R. Its Stieltjes transform at spectral parameter z ∈ H is given by Z dµ(x) mµ(z) := R x − z Easy facts:

• z → mµ(z) is analytic in H with image in H: Z dµ(x) Im ( ) = = + m z η 2 2, z E iη R |x − E| + η

• iη mµ(iη) → −1 as η → ∞ •| m(z)| ≤ 1 ; Im z • These 3 properties characterize the Stieltjes transform (i.e. for any such function m(z) there is a prob. measure s.t. m = mµ).

31 Message: mµ(E + iη) resolves the measure µ around E on scale η 1 Z Im mµ(z) := (hη ? µ)(E) = hη(x − E)dµ(x) π R where hµ is an approximate delta fn. on scale η 1 η Z hµ(x) := , hµ(x)dx = 1 π x2 + η2

Inversion formula holds 1 lim Im mµ(E + iη) = µ(E)dE η→0+ π (weak convergence).

Similarly to the Fourier transform, the pointwise convergence of the Stieltjes tr. characterizes weak convergence of prob. measures:

µN * µ iﬀ mµN (z) → mµ(z) ∀z ∈ H More precise results on speed of convergence are also available. 32 Resolvents and Stieltjes transform

Obvious fact: The trace of the resolvent G of a hermitian matrix H is the Stieltjes transform of its empirical spectral density: N 1 X 1 1 X 1 %N (E) := δ(λα − E), TrG(z) = = m%N (z) N α=1 N N α λα − z Clearly the limit

lim Im m% (E + iη) η→0+ N may not exist, but its expectation may exist

lim E Im m% (E + iη) = %(E) η→0+ N and gives the density of states (DOS). Moreover, even without expectation, we can hope that 1 m (E + iη) ≈ m%(E), η N N holds with very high probability. This tells us that the eigenvalues are uniformly distributed down to the smallest possible scales η 1/N.

33 Summary up to now

— We study eigenvalue statistics of large N × N random matrices.

— Wigner matrices: i.i.d. entries vs. invariant ensembles.

— Normalization: λj ∼ O(1), gap = λi+1 − λi ∼ 1/N (bulk)

— Scales: Global [O(1)], Mesoscopic, Microscopic O(1/N).

— Wigner’s fundamental vision: Global density may be model dependent, but microscopic statistics is universal, depends only on the symmetry type [Wigner-Dyson-Mehta universality]

— WDM universality has recently been proven for Wigner matrices via the three-step strategy. Main input: local law on optimal scale.

34 — Resolvent has good properties: for small η = Im z it resolves the empirical density of eigenvalues on scale η and it is self-averaging (LLN) if η 1/N

— Stieltjes transform of a measure Z dµ(x) mµ(z) := R x − z

— empirical density 1 P δ is close to the semicircle on (short) N j λj scale η 1/N if the corresponding Stieltjes transforms are close 1 TrG(z) ∼ msc(z) N

— Even better, G(z) ≈ M(z) (resolvent is self-averaging) where

−M−1 = z − A + S[M], Im M > 0 h i with A = EH, S[R] := E (H − A)R(H − A) .

35 Models of increasing complexity

2 1 • Wigner matrix: i.i.d. entries, sij := E|hij| are constant (= N ). (Density = semicircle; G ≈ diagonal, Gxx ≈ Gyy) [E-Schlein-Yau-Yin, 2009–2011], [Tao-Vu, 2009]

P • Generalized Wigner matrix: indep. entries, j sij = 1 for all i. (Density = semicircle; G ≈ diagonal, Gxx ≈ Gyy) [E-Yau-Yin, 2011], [E-Knowles-Yau-Yin, 2012]

• Wigner type matrix: indep. entries, sij arbitrary (Density 6= semicircle; G ≈ diagonal, Gxx 6≈ Gyy) [Ajanki-E-Kr¨uger,2015]

• Correlated Wigner matrix: correlated entries, sij arbitrary (Density 6= semicircle; G 6≈ diagonal) [Ajanki-E-Krüger’15-’16] [Che ’16], [E-Krüger-Schröder’17], [Alt-E-Krüger-Schröder’18]

36 Other extensions of the original Wigner model (SKIP)

h i • Invariant ensembles: P (H) ∼ exp − βNTrV (H) Deift et. al., Pastur-Shcherbina, Bourgade-E-Yau, Bekerman-Guionnet-Figalli, etc.

• Low moment assumptions, heavy tails Johansson, Guionnet-Bordenave, G¨otze-Naumov-Tikhomirov,Benaych-Peche, Aggarwal

• Deformed models, general expectation O’Rourke-Vu, Lee-Schnelli-Stetler-Yau, He-Knowles-Rosenthal

• Sparse matrices, Erd˝os-R´enyiand d-regular graphs E-Knowles-Yau-Yin, Huang-Landon-Yau, Bauerschmidt-Huang-Knowles-Yau, etc.

• Band matrices Fyodorov-Mirlin, Disertori-Pinson-Spencer, Schenker, Sodin, E-Knowles-Yau, T. Shcherbina, Bourgade-E-Yau-Yin, E-Bao etc.

Many other directions and references are left out, apologies... 37 OVERVIEW OF RESULTS

• Properties of the limiting (self-consistent) density of states (DOS), special focus on their singularity structure;

• Local laws, i.e. approximating G(z) with a deterministic quantity M(z) down to the smallest possible scale η = Im z 1/N;

• Universality of the eigenvalue statistics on the scale 1/N.

Most results will be presented only informally; the precise statement of the local law will be given later.

First some pictures of the DOS for Wigner type matrices:

38 Variance proﬁle and self-consistent density of states (DOS)

0.30

0.25

0.20

0.15

0.10

X 0.05

0.00 sij = 1 ⇐⇒ -2 -1 0 1 2 j 2 General variance proﬁle sij = E|hij| : not the semicircle any more.

P j sij 6= const= ⇒ Density of states 39 Features of the DOS for Wigner-type matrices

Support splits via cusps:

(Matrices in the pictures represent the variance matrix)

40 Universality of the DOS singularities

√ Edge, E singularity Cusp, |E|1/3 singularity

Small-gap Smoothed cusp √ 1+ 2 √ (2+τ)τ √ √ √τ − 1 1+(1+τ+ (2+τ)τ)2/3+(1+τ− (2+τ)τ)2/3 ( 1+τ 2+τ)2/3+( 1+τ 2−τ)2/3−1

τ := |E| , τ := |E| gap (minimum )1/3

41 Main theorems on shape analysis of DOS

Recall the (matrix) Dyson equation, for any parameter z ∈ C+,

1 N×N − = z − A + S[M],M = M(z) ∈ C , Im M > 0 M ∗ N×N N×N where A = A and S : C → C is symm., positivity preserving.

The corresponding DOS, ρ(x) is characterized via 1 Z ρ(x)dx TrM(z) = N x − z

Suppose ﬂatness (for simplicity), i.e. for any positive matrix R ≥ 0 c C TrR ≤ S[R] ≤ TrR N N In this setup, we have the following properties of the DOS

42 Theorem [Ajanki-E-Kr¨uger,2016][Alt-E-Kr¨uger,2018]

• ρ(x) is real analytic on the set {ρ > 0};

1 • ρ(x) is 3-H¨oldercontinuous in R

• The singularities of ρ, wherever it vanishes, are universal:

q – ρ(x) ∼ (x − E)+ near a right edge

– ρ(x) ∼ |x − E|1/3 at a cusp

– ρ(x) ≈ explicit functions near ”almost-cusps”

• No other singularity may occur (somewhat surprising)

43 Main theorems on local laws and universality (informally)

Theorem [Ajanki-E-Kr¨uger,2014] Let H = H∗ be a Wigner-type matrix with general variance proﬁle c/N ≤ sij ≤ C/N. Then optimal local law (including edge) and bulk universality hold.

Theorem [Ajanki-E-Krüger,2016, E-Krüger-Schröder2017] Let H = H∗ be correlated 1 H = A + √ W N where A is deterministic, W is random with EW = 0 and polynomial decay of correlation. Assume the lower bound

∗ 2 2 2 E|u W v| ≥ ckuk kvk ∀u, v. Then optimal local law and bulk universality hold.

44 Polynomial decay of correlation (SKIP)

Distance for index sets in the condition on the correlation decay

C(φ, ψ) Cov φ(WA), ψ(WB) ≤ h is 1 + dist(A, B) for any A, B ⊂ S × S assumes the usual metric on the set S = {1, 2,...,N} of indices. Here WA = {Wij :(i, j) ∈ A}.

d(A, B) A A B ( B ) 45 Local law and ”usual” corollaries

There exists a deterministic matrix M, with kMk = O(1) s.t. 1 Gij(z) − Mij(z) . √ NIm z

1 1 kBk TrBG(z) − TrBM(z) N N . Nη with very high prob.

• Delocalization of bulk eigenvectors • Rigidity of bulk eigenvalues • Wigner-Dyson-Mehta universality in the bulk

Remark: Very recently all the same results have been extended to the edge (Tracy-Widom universal statistics) and to the cusp [Landon-Yau], [Alt-E-Krüger-Schröder],[E-Krüger-Schröder]

46 Delocalization

Let u be a bulk eigenvector, Hu = λu, %(λ) ≥ c > 0, then Nε max |u(i)| ≤ √ kuk i N for any ε > 0 with very high probability. Shows that the system is in the delocalized regime.

2 Ex: Prove |u(x)| ≤ η maxE Im Gxx(E + iη) via spectral theorem. Rigidity

For any E, let k(E) be the index of the corresponding quantile, i.e. Z E k(E) := N %(x)dx −∞ Then for any bulk energy, %(E) ≥ c > 0, Nε λ − E ≤ k(E) N with very high probability, i.e. eigenvalues rigidly stick. 47 Bulk universality: sine-kernel statistics holds on the level of individual eigenvalues:

2.0

1.0

- 3.0 - 2.0 - 1.0 - 0.0

48 Universality conjecture for disordered quantum systems:

A disordered quantum system with suﬃcient complexity exhibits one of the following two behaviors:

d • Insulator (typically at strong disorder; for simplicity Σ = Z ) – Localized eigenvectors, i.e. ∃Σ0 ⊂ Σ, |Σ0| |Σ| s.t.

X |ψ(x)|2 1. x∈Σ\Σ0 – exp. oﬀdiag decay of the resolvent

−|x−x0|/` |Gxx0| ≤ Ce , ` localization length – lack of transport

X 2 2 itH sup x |ψt(x)| ≤ C < ∞; ψt = e ψ0 t∈R x – nearby eigenvalues are independent (Poisson local spectral statistics).

49 • Conductor (typically at weak disorder) – Delocalized eigenvectors – non-integrable decay of the resolvent – transport (via quantum diﬀusion) – eigenvalues are strongly correlated (Wigner-Dyson local statistics)

At ﬁrst sight, localization is surprising (Anderson). Still, mathemat- ically it is more accessible (Fr¨ohlich-Spencer, Aizenman-Molchanov, Minami, ...).

Anderson metal-insulator quantum phase transition

50 Two popular models to study the dichotomy

d d (1) Random Schr¨odingeroperators: in lattice box Σ := [1,L] ∩ Z

In d = 1 it corresponds to a narrow band matrix with i.i.d. diagonal:   v1 1    1 v2 1   .   1 ..  H =    ... 1     1 1   vL−1  1 vL

Follows insulator behavior 51 (2) Mean ﬁeld Wigner random matrices: On the set Σ = {1, 2,...,N}

∗ H = (hxy),H = H Ehxy = 0. entries are identically distributed and independent up to symmetry.

  h11 h12 . . . h1N    h21 h22 . . . h2N  H =  . . .   . . .  hN1 hN2 . . . hNN

H models a mean-ﬁeld hopping mechanism with random quantum transition rates. No spatial structure.

Follows conductor behavior

52 Band matrices: interpolation between Anderson and Wigner

d d Σ = [1,L] ∩ Z lattice box Entries of H = H∗ are indep but not identically distr. Wigner type! ∗∗∗ 0 0 0 0 ∗∗∗∗ 0 0 0 2 1 |x − y| ∗∗∗∗∗ 0 0 E|hxy| = f = 0 ∗∗∗∗∗ 0 W d W H   0 0 ∗∗∗∗∗  0 0 0 ∗∗∗∗ W is the bandwidth (interaction 0 0 0 0 ∗∗∗ range) Band matrix in d = 1 physical dim.

W = O(1)[ ∼ Anderson] ←→ W = L [Wigner]

”Facts” from physics: Transition occurs at W ∼ L1/2 (d = 1) SUSY [Fyodorov-Mirlin, 1991] W ∼ plog L (d = 2) RG scaling [Abrahams, 1979] W ∼ W0(d)(d ≥ 3) extended states conjecture

53 Mean ﬁeld quantum Hamiltonian with correlation

Equip the conﬁguration space Σ with a metric to have ”nearby” states.

It is reasonable to allow that hxy and hxy0 are correlated if y and y0 are close with a decaying correlation as dist(y, y0) increases.

Non-trivial spatial structure changes the density of states.

There are many other natural models leading to correlated struc- tures. Now we turn to some proofs. 54 Some proof ideas.

There are two basic methods to study RM eigenvalue statistics, i.e. 1 P for understanding the empirical ev. measure µN (dx) = N i δ(λi −x)

1 k R k – Moment method. Find N ETrH = E x µN (dx).

– Resolvent method: Use that resolvent = St. transform of µN (dx) 1 1 1 Z µ (dx) TrG(z) = Tr = N , z = E + iη N N H − z R x − z and use many convenient properties of the resolvent.

Moment method is suitable for global laws and for extreme edges, even for complicated models involving many matrices or free probability, but ineﬃcient for local information inside the spectrum.

We will follow the resolvent method and we ﬁrst informally present it for Wigner type matrices. 55 Def: [Wigner type matrix] H is an N × N hermitian random matrix 2 Ehij = 0, sij := E|hij| In this case M = diag(m) diagonal and the Dyson eq. becomes:

Def: [Quadratic vector equation] Given z ∈ H := {Im z > 0}, consider 1 − = z + (Sm)i, i = 1, 2,...N, [QV E] mi

The QVE has a (unique) solution in the upper half plane denoted N by m(z) = (m1(z), . . . mN (z)) ∈ H .

1 Key relation [informally] G = H−z is close to M = diag(m);

Gii(z) ≈ mi(z),Gij(z) ≈ 0, (i 6= j) N → ∞

56 Structure of the proof of the local law

Step 1: Probabilistic step (derivation of the QVE)

Prove that gi := Gii approximately satisﬁes the QVE, i.e. 1 − = z + (Sg)i + di, (∗) gi for some small (random) error vector d = (di). ⇒ Exercise session

Step 2: Deterministic step (Stability of the QVE)

Consider (*) as a small perturbation of the QVE: 1 − = z + (Sm)i and (∗) =⇒ km − gk . kdk mi

Key question: stability of the QVE in an appropriate norm/space.

Tool: analyse the linear stability operator (1 − m2S)−1. 57 RECAPITULATION: Three-step strategy.

1. Local density law down to scales 1/N

(Needed in entry-wise form, i.e. control also matrix elements Gij the resolvent G(z) = (H − z)−1 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.

Step 1 is model dependent and has been the main topic today.

Some short comments on Step 2 and 3.

58 Step 2: Dyson Brownian Motion

Gaussian convolution matrix interpolates between Wigner and GUE.

H = H0 −→ Ht −→ H∞ = GUE Embed H into an Ornstein-Uhlenbeck matrix ﬂow: 1 1 −t/2 −t 1/2 dHt = √ dBt − Htdt Ht ∼ e H0 + (1 − e ) V. N 2

Dyson 1 1 1 1 =⇒ dλ = √ dB + − λ + X dt i i 2 i N N j6=i λi − λj Ev-ﬂow becomes a stochastic dynamics of interacting “particles”.

Idea: Equilibrium is GUE/GOE with known local statistics.

Global equilibrium is reached in time O(1) (convexity, Bakry-Emery).

For local statistics, only local equilibrium needs to be achieved which is much faster. The main result proves Dyson’s conjecture: [E-Schlein-Yau] [E-Schlein-Yin-Yau] [Landon-Yau] [E-Schnelli] [Landon-Sosoe-Yau]

59 “The picture of the gas coming into equilibrium in two well- separated stages, with microscopic and macroscopic time scales, is suggested with the help of physical intuition. A rigorous proof that this picture is accurate would require a much deeper mathematical analysis.”

Freeman Dyson, 1962

on the approach to equilibrium of Dyson Brownian Motion

Global equilibrium is reached in time scale of O(1). Local equilibrium was believed to be reached in O(N−1).

Thm. Local law on scale η N−1 implies local eq. in time O(N−1). 60 Time scales for DBM

H GUE

Resolvent perturbation regime

time -1 -ε 0 N N 1 Global eq. Local equilibrium

61 Step 3: Resolvent perturbation with four moment matching

Green Function Comparison Thm. Suppose H and Hc have matching ﬁrst four moments and entrywise local law is known for both. Then expectations of resolvents with Im z slightly below 1/N are close:

h −1−ε i h −1−ε i Φ TrG(E + iN ) − Φ TrGb(E + iN ) kΦk E E where Φ is a smooth fn, possibly with several G’s in the argument.

This can detect local statistics of individual eigenvalues.

Proof. Replace the entries of H to Hc one by one by res. expansion:

G(h12) = G0 + G0h12G0 + G0h12G0h12G0... G0 := G(h12 := 0)

G(hb12) = G0 + G0hb12G0 + G0hb12G0hb12G0... G0 := G(h12 := 0)

Using G0 is indep of h12, hb12, ﬁrst four terms are the same in expectation. Fifth order term is N−5/2, combinatorics N2 – small! (For general Φ use Taylor) [Similar idea: Tao-Vu ]

62 SUMMARY

• Wigner’s vision: universality of local ev. stat.

• Three step strategy: local law is model dependent

• We reviewed local laws for various random matrix ensembles. Dyson eq. and its stability plays a central role.

• Dyson Brownian motion is the mechanism for univ.

• For correlated random matrices we proved optimal local law and bulk universality.

• Very recently the theory is extended to the edge and the cusp