Universality for random matrices and log-gases

“Broadening the knowledge base and supporting the long term professional sustainability of the Research University Centre of Excellence

at the University of Szeged by ensuring the rising generation of excellent scientists.””

Doctoral School of Mathematics and Computer Science

Stochastic Days in Szeged 26.07.2012.

Universality for

random matrices and log gases László Erdős

(Ludwig-Maximilians-Univeristät München)

Universality for random matrices and log-gases

L´aszl´o Erd˝os IST, Austria

Ludwig-Maximilians-Universit¨at, Munich, Germany Conference for Andr´as Kr´amli’s 70-th birthday

University of Szeged, Jul 26-27, 2013

With P. Bourgade, B. Schlein, H.T. Yau, and J. Yin

Eugene Wigner (1956):

Successive energy levels of large nuclei have universal spacing statis- tics that can be described by random matrices. [Physics]

Freeman Dyson (1962):

A gas of particles with a logarithmic interaction reaches local equi- librium very fast. [Statistical Mechanics]

De Giorgi-Nash-Moser (1957-60):

Uniformly elliptic and parabolic equations in divergence form with rough coefficients have H¨older continuous solutions. [Math]

What do these facts have in common?

RANDOM MATRICES

Basic question [Wigner]: Is there some universal pattern in the eigenvalues of large random matrices?

H =











h₁₁ h₁₂ . . . h_1N h₂₁ h₂₂ . . . h_2N

... ... ...

h_N1 h_N2 . . . h_{N N}











=⇒ (λ₁, λ₂, . . . , λ_N) eigenvalues?

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

Analogy: Central limit theorem: ^√1

N(X₁ +X₂ +. . .+X_N) ∼ N(0, σ²)

Gaussian Unitary Ensemble (GUE):

H = (h_jk) complex hermitian N × N matrix

h_jk = ¯h_kj (for j < k) are complex and h_kk are real independent Gaussian random variables with normalization

Eh_jk = 0, E|h_jk|² = 1 N.

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

1 N

X i

λ²_i = E 1

NTrH² = 1 N

X ij

E|h_ij|² = 1

(Similar for GOE, GSE)

Wigner’s observations

i) Density of eigenvalues:

Wigner semicircle law

−2 2

ρ 2π

1

(x) = 4 − x2

ii) Level repulsion: Wigner surmise (in the bulk and for GOE) P

λ_i+1 − λ_i = s N

≈ πs

2 exp

− π 4s²

ds

Guessed by a 2x2 matrix calculation

SINE KERNEL FOR CORRELATION FUNCTIONS

Probability density of the eigenvalues: p(x₁, x₂, . . . , x_N) The k-point correlation function is given by

p^(k)_N (x₁, x₂, . . . , x_k) :=

Z

R^N−k

p(x₁, . . . x_k, x_k+1, . . . , x_N)dx_k+1. . . dx_N

Special case: k = 1 (density) is used to compute one-point observ- ables

E 1 N

N X i=1

O(λ_i) =

Z

O(x)p⁽¹⁾_N (x)dx → 1 2π

Z

O(x)

q

4 − x²dx

p^(k)_N with higher k computes observables with k evalues.

Rescaled correlation functions at energy E p^(k)_E (x^{) := 1}

[%(E)]^kp^(k)_N

E + x₁

N %(E), E + x₂

N %(E), . . . , E + x_k N %(E)

Rescales the gap λ_i+1 − λ_i to O(1).

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

lim

N→∞p^(k)_E (x^{) = detn}S(x_i − x_j)^ok

i,j=1, S(x) := sinπx πx

p⁽²⁾_E (x⁾ → 1 −

sinπ(x₁ − x₂) π(x₁ − x₂)

₂

(=⇒ Level repulsion)

The limit is independent of E as long as |E| < 2 (bulk spectrum)

Main question: going beyond Gaussian towards universality!

There are two disjoint directions of generalization: the Gaussian is the common intersection.

MODEL 1: INVARIANT ENSEMBLES (LOG-GAS)

Unitary ensemble: Hermitian matrices with density P(H)dH ∼ e⁻TrV (H)

dH

Invariant under H → U HU⁻¹ for any unitary U

Joint density function of the eigenvalues is explicitly known p(λ₁, . . . , λ_N) = const. ^Y

i<j

(λ_i − λ_j)^βe⁻

P

j V (λ_j)

classical ensembles β = 1,2,4 (orthogonal, unitary, symplectic sym- metry classes; GOU, GUE, GSE for Gaussian case, V (x) = x²/2) General β > 0: Gibbs measure with inv. temp. β (no matrix):

Y i<j

(λ_i−λ_j)^βe^{−βN Pi V (λi)} ∼ e^−βNH(λ), H = ^X

i

V (λ_i)− 1 N

X i<j

log(λ_j−λ_i) Prototype of strongly correlated stat.mech. system (“log-gas”).

Universality conjecture for log-gases:

For any β > 0, the local statistics is independent of V

For classical β = 1,2,4, correlation fn’s can be explicitly computed via large N asymptotics of orthogonal polynomials due to the Van- dermonde determinant.

Dyson, Gaudin, Mehta: 60’-70’s. Gaussian case (Hermite poly)

Deift, Pastur-Shcherbina, Bleher-Its, Lubinsky: from 90’s, general case

For general β? New method is needed!

Main goal: develop new methods in statistical mechanics of strongly correlated systems.

MODEL 2: (GENERAL) WIGNER ENSEMBLES

H = (h_ij)_1≤i,j≤N, ¯h_ji = h_ij independent Eh_ij = 0, E|h_ij|² = s_ij, ^X

i

s_ij = 1,

c

N ≤ s_ij ≤ C N

If h_ij are i.i.d. then it is called Wigner ensemble.

Universality conjecture (Dyson, Wigner, Mehta etc) : If h_ij are independent, then the local eigenvalue statistics are the same as for the Gaussian ensembles. Only symmetry type matters.

No previous results (apart from Johansson’s for hermitian matrices

SOLUTION TO THE UNIVERSALITY CONJECTURES

Theorem [E-Schlein-Yau-Yin, 2009-2010] Local ev. statistics is universal for generalized Wigner ensembles in the bulk (and edge).

[Tao-Vu, 2010] Hermitian case via moment matching.

Theorem [Bourgade-E-Yau, 2011-2013] Let β > 0 and V ∈ C⁴. Then local statistics is universal in the bulk.

Formally: weak convergence of the rescaled correlation functions p^(k)_E (x⁾ → universal as N → ∞

Key ingredient: Uniqueness of the local Gibbs state with log-interaction What else is left? Single gap – subtle difference from corr. fn’s.

UNIVERSALITY OF GAPS

Theorem [E-Yau, 2012] Let β ≥ 1 and V real analytic. Then single gap for the log-gas is universal in the bulk.

Precise formulation: Let µ_V and µ_G denote the β-log-gas with V and the Gaussian case. Then

E^µV ON(λ_i+1 − λ_i) − E^µGON(λ_j+1 − λ_j)

≤ CN^−ε for any fixed label i, j in the bulk.

ε is a H¨older regularity exponent from De Giorgi-Nash-Moser theory!

Theorem [E-Yau, 2012] The single-gap universality holds for gen- eralized Wigner ensembles in the bulk.

Previous result by Tao [2012] for GUE (explicit formulas) + hermi-

MAIN STEPS IN THE PROOF

i) Localization of the problem

ii) Representation as a random walk with long ranged jumps.

iii) Rigidity and level repulsion for the local equilibrium measure

iv) De Giorgi-Nash-Moser estimate with L¹ upper bound on the data

LOCAL MEASURE

Fix an interval of indices I = [L − K, L + K] and rename the points (λ₁, λ₂, . . . , λ_N) := (y₁, . . . y_L−K−1, x_L−K, . . . , x_L+K, y_L+K+1, . . . y_N)

00 11

00 11

00 11

00 11

00 11

00 11

00 11

x y

Configurational interval Jy

y

"boundary conditions"

"boundary conditions"

Given y fixed, define the conditional measure µ_y(dx). It is again a β-log-gas, but with non-smooth potential and in J_y.

µ_y ∼ e^{−N βHy}, H^y(x) := X

i∈I

1

2V_y(x_i) − 1 N

X

i<j∈I

log|x_j − x_i|

V_y(x) := V (x) − 2 N

X

k6∈I

log|x − y_k| (external potential)

GAP UNIVERSALITY FOR THE LOCAL MEASURE

Take two β-log-gases with different potentials and localize them at different intervals. Assume the boundary conditions are “regular”

Theorem: [Informally] The local gaps have the same distribution.

L+q L+K+1

L

~ ~

~

ρV ρ

V

~

~

q−th gaps are compared

INTERPOLATION BETWEEN µ AND µ_e

Given µ = µ_y,V and µ_e = µ

ye,Ve, interpolate between them by defining ω_y^r_,

ye := e^−βrh(x)µ_y, h(x^{) :=} NV^e

ye(x⁾ − V_y(x⁾, r ∈ [0,1]

=⇒

h

E^µ − E^µe

iO(x_q − x_q+1)

= β

Z ₁

0 dr hh; O(. . .)i_ω^r

hf; gi = E^ωf g − E^ωfE^ωg is the covariance.

Need to show that

hh(x^);O(x_q − x_q+1)i_ω ≤ K^−ε. (1) This is a statement about the decay of the “point-gap” covariance.

COVARIANCE STRUCTURE OF LOG-GASES

If µ is a log-gas on K points with interaction ^P_i<j log(x_j − x_i), then hx_i;x_ji_µ ∼ log K

h|i − j| + 1ⁱ

(2)

“point-point” covariance decays slowly. But “point-gap” is better:

hx_i ; x_j − x_j+1i_µ ∼ d

dj log K

h|i − j| + 1ⁱ ∼ 1

|i − j| (3) Only (2) is proved for GUE using explicit formulas [Gustaffson], the rest are conjectures!

Our main result (1) essentially proves (3) with |i − j|^−ε.

Usual methods for Gibbs measures do not apply due to the very strong correlation. Lack of methods!

RANDOM WALK REPRESENTATION

Let ω = e^−H be any Gibbs measure and L = ∆− ∇H· ∇ its generator and x(t) is the stochastic process generated by L, i.e.

dx(t) = dB(t) − ∇H(x(t))dt

Representation formula for the point-gap covariance:

hh(x^);O(x_q−x_q+1)i_ω =

Z _∞ 0 dσ

Z

ω(dx⁾_Ex^hw_q(σ)−w_q+1(σ)ⁱO⁰(x_q−x_q+1)

where w(s) = w(s, x(·)) (depending on the whole path) solves

∂_sw(s) = −A(s)w(s), A(s) = H⁰⁰(x(σ−s)), w^{(0) =} ∇h(x(σ)).

Similar formulas by [Helffer-Sj¨ostrand] and [Naddaf-Spencer] and ...

Note that w_q − w_q+1 is a discrete derivative!!

DE GIORGI-NASH-MOSER EMERGES

hh(x^);O(x_q−x_q+1)i_ω =

Z _∞ 0 dσ

Z

ω(dx⁾_Ex^hw_q(σ)−w_q+1(σ)ⁱO⁰(x_q−x_q+1)

∂_sw^{(s) =} −A(s)w^(s),

[A(s)w^]_j ⁼ −^X

i

Bij(s)(w_i−w_j)+potential, Bij(s) = 1

(x_i(s) − x_j(s))² If x_i(s) were regularly spaced, then Bij(s) ∼ (i − j)², i.e. the elliptic part B were the discretization of |p| = √

−∆, and we had for the derivative of the fundamental solution

Z _∞ 0 dσ

e^−σ|p|

q,j

−

e^−σ|p|

q+1,j

= ∇_q

1

|p|

j,q

∼ 1

|q − j|

Adapting De Giorgi-Nash-Moser, in a recent paper Caffarelli et. al.

proved H¨older regularity for a continuous version of this equation, where Bij was replaced with B(x − y) ∼ |x − y|⁻².

Lower bound (for ellipticity) on Bij = ¹

(x_i−x_j)² follows from |x_i−x_j| . K^ξ|i − j| which is easy to guarantee by apriori location bound

Upper bound (needed in De Giorgi-Nash-Moser) is critical:

Bi,i+1(s) ≤ C ⇐⇒ |x_i(s) − x_i+1(s)| ≥ C⁻¹ (4) Level repulsion has only polynomial probability

P^ω(|x_i(s) − x_i+1(s)| ≤ ε) ∼ ε^β

so (4) cannot be guaranteed simultaneously for all s and i.

We develop De Giorgi-Nash-Moser with L¹ input on Bij.

SUMMARY

1. We proved bulk (and edge) universality for β-log-gases

2. We proved bulk (and edge) universality for generalized Wigner matrices (Wigner-Dyson-Mehta conjecture)

3. In both models we proved universality in both senses; averaged- energy and fixed gap.

4. We established the uniqueness of the Gibbs state for log-gases

OPEN QUESTIONS

1. Prove fixed energy universality beyond Hermitian case

2. Depart from mean field models via band matrices towards ran- dom Schr¨odinger

3. Understand general properties of log-gases, as a universal strongly correlated system.

BOLDOG SZ ¨ ULINAPOT ANDR ´ AS !!

Мой стих трудом

громаду лет прорвет и явится

весомо,

грубо,

зримо как в наши дни

вошел водопровод, сработанный

еще рабами Рима.

[M., B. B.]

hh(x^);O(x_q−x_q+1)i_ω =

Z _∞ 0 dσ

Z

ω(dx⁾_Ex^hw_q(σ)−w_q+1(σ)ⁱO⁰(x_q−x_q+1)

∂_sw(s) = −A(s)w(s), A(s) = H⁰⁰(x(σ − s)), w_j(0) = ∇_jh

“Proof:” Using the commutator [L,∇] = ∇ · H⁰⁰∇, the covariance is hf ; gi_ω = (f,L1

Lg)_ω = (∇f, ∇1

Lg)_ω = ∇f, 1

L + H⁰⁰∇g

ω

Since x^(t) is generated by L, then by Feynman-Kac:

hf ; gi_ω

Z _∞

0 dσ∇f(x(σ)), e^−σH00∇g(x⁽⁰⁾⁾

ω

Not quite true, since H⁰⁰ = H(x(t)) is time dependent.

Correctly: use the time dependent version of Feynman-Kac.