Optimal global rigidity estimates in unitary invariant ensembles - - PowerPoint PPT Presentation

Optimal global rigidity estimates in unitary invariant ensembles

Tom Claeys joint work with Benjamin Fahs, Gaultier Lambert and Christian Webb Integrability and Randomness in Mathematical Physics and Geometry CIRM Luminy, April 8, 2019

Global rigidity of eigenvalues

Random matrix eigenvalues Fundamental question in random matrix theory is to understand eigenvalue statistics of large random matrices

distribution, macroscopic linear statistics ...

correlations, extreme eigenvalue distribution

their classical positions

Global rigidity in the GUE

Classical GUE eigenvalue locations Let be the eigenvalues of a GUE matrix

• f size

, normalized such that the eigenvalue distribution converges to a semi-circle law on . (Equivalently, is Hermitian and the independent entries are iid (real

• n the diagonal, complex otherwise) Gaussians with variance

.) Classical locations are given by

≤ ≤ ⋯ ≤ λ1 λ2 λN M N × N [−1,1] M Mi,j

1 4n

,…, ∈ [−1,1] κ1 κN dx = .

2 π ∫ κj −1

1 − x2 − − − − − √

j N

Global rigidity in the GUE

Global rigidity What can we say for large about the distribution of the normalized maximal fluctuation of eigenvalues

N := { | − |}? MN max

j=1,…,N

2 π 1 − κ2

j

− − − − − √ λj κj

Global rigidity in the GUE

Upper bound for generalized Wigner matrices (ERDOS-YAU-YIN '12 ) Lower bound for GUE (GUSTAVSSON '05) for , which implies (non-optimal) lower bounds for .

P( ≥ ) ≤ C exp(−c(logN ) MN (logN)α log log N N )

log log N α′

2 ( − ) → N (0,1) 2 – √ 1 − κ2

j

− − − − − √ N logN − − − − − √ λj κj δ ≤ j ≤ (1 − δ)N MN

Global rigidity in the GUE

Theorem (C-Fahs-Lambert-Webb '18) For any , we have Unitary invariant ensembles A similar result holds for unitary invariant ensembles with eigenvalue distribution for real analytic with sufficient growth at .

ϵ > 0 P((1 − ϵ) < < (1 + ϵ) ) = 1. lim

N→∞

logN πN MN logN πN − d 1 ZN ∏

1≤i<j≤N

∣ ∣λi λj∣ ∣

2 ∏ 1≤j≤N

e−NV(

) λj

λj V ±∞

Global rigidity in unitary invariant ensembles

Equilibrium measure and classical locations Semi-circle law is then replaced by the equilibrium measure minimizing We assume that is one-cut regular, and that the support is for convenience. The classical locations are now defined by

μV log|x − y dμ(x)dμ(y) + V (x)dμ(x). ∫

R×R

|−1 ∫

R

μV [−1,1] ,…, ∈ [−1,1] κ1 κN d (x) = . ∫ κj

−1 μV j N

Global rigidity in unitary invariant ensembles

Theorem (C-Fahs-Lambert-Webb '18) For any , we have

ϵ > 0 P( < max{ ( )| − |} < ) = 1. lim

N→∞

(1 − ϵ)logN πN dμV dx κj λj κj (1 + ϵ)logN πN

Global rigidity in unitary invariant ensembles

Eigenvalue counting function We prove this via the extrema of the normalized eigenvalue counting function Namely, we prove that for any , Heuristically, we expect which explains the connection between global rigidity and the maximum of the normalized eigenvalue counting function.

(x) = π( − N d ), x ∈ R. hN 2 – √ ∑

1≤j≤N

1

≤x λj

∫

x −1

μV δ > 0 P[(1 − δ) logN ≤ { ± (x)} ≤ (1 + δ) logN] = 1. lim

N→∞

2 – √ max

x∈R

hN 2 – √ ( ) = d (x) ≈ ( )( − ), hN λj ∫ κj

λj

μV

dμV dx κj

κj λj

Extreme values of the eigenvalue counting function

Extreme of log-correlated fields behaves for large like a stochastic process with log-correlations (JOHANSSON '98) How to estimate extrema of log-correlated processes? This question has been studied in different contexts.

ARGUIN-BELIUS-BOURGADE '16, CHHAIBI-MADAULE-NAJNUDEL '16)

MADAULE-NAJNUDEL '16, PAQUETTE-ZEITOUNI '16, HOLCOMB- PAQUETTE '18)

ensembles (FYODOROV-SIMM '14, LAMBERT-PAQUETTE '18)

hN N ζ

Extreme values of the eigenvalue counting function

Multiplicative chaos Powerful tools to study such extrema come from the theory of multiplicative chaos

'15)

'14, WEBB '15, BERESTYCKI-WEBB-WONG '18, LAMBERT-OSTROVSKY- SIMM '18) Exponential moments Crucial input for this method: good control of exponential moments and for large

Eeγ

(x) hN

Ee

( )+ ( ) γ1hN x1 γ2hN x2

N

Extreme values of the eigenvalue counting function

Upper bound estimates Upper bound for can be obtained using an elementary

• ne-moment method.
• 2. By a union bound and Markov's inequality,
• 3. Substitute large

asymptotics for and choose as big as possible such that rhs decays for some .

{ ± (x)} maxx∈I hN

1.

{ ± (x)} ≤ { ± ( )} + 1. max

x∈I

hN max

j: ∈I κj

hN κj P( { ( )} > Y ) ≤ P( ( ) > Y ) ≤ . max

j: ∈I κj

hN κj ∑

j: ∈I κj

hN κj ∑

j: ∈I κj

Eeγ

( ) hN κj

eγY N Eeγ

(x) hN

Y γ

Extreme values of the eigenvalue counting function

Upper bound estimates is a Hankel determinant with discontinuous weight , and large asymptotics for such Hankel determinants are known for (ITS-KRASOVSKY '08 for GUE, CHARLIER '18 for one-cut regular unitary invariant ensembles): To extend this to all eigenvalues, we need a similar result for close to . We prove

Eeγ

(x) hN

e−NV(λ)eγ1λ≤x N x ∈ (−1 + δ,1 − δ) E ≤ , x ∈ (−1 + δ,1 − δ). eγ

(x) hN

CγN

γ2 2

x ±1 E ≤ (1 − , |x| ≤ 1 − m . eγ

(x) hN

C ′

γN

γ2 2

x2)

3γ2 4

N −2/3

Extreme values of the eigenvalue counting function

Lower bound estimates Optimal lower bound estimates are much harder to obtain, and require to investigate the log-correlated structure of . Log-correlated structure behaves for large (JOHANSSON '98) like a Gaussian process with logarithmic covariance kernel

hN hN N X(x) Σ(x,y) := log . ∣ ∣ ∣ 1 − xy + 1 − x2 − − − − − √ 1 − y2 − − − − − √ x − y ∣ ∣ ∣

Multiplicative chaos

Maximum of the eigenvalue counting function For studying the maximum of , we prove that the random measure converges weakly in distribution to a multiplicative chaos measure which can be formally written as (cf. KAHANE '85, RHODES-VARGAS '10, BERESTYCKI '17, BERESTYCKI-WEBB-WONG '17)

hN d = dx, γ ∈ R μγ

N

eγ

(x) hN

Eeγ

(x) hN

d (x) = dx. μγ eγX(x) EeγX(x)

Multiplicative chaos

Extreme values It will turn out that the extreme values of the limiting measure will lead us to estimates for extreme values of . Heuristics Heuristically, the random measure is expected to be dominated for by -values where is exceptionally large, namely and it is natural to expect that the multiplicative chaos measure will give us information about large values of . For , , which suggests heuristically that values where are unlikely to occur.

μγ hN d (x) = dx μγ

N eγ

(x) hN

Eeγ

(x) hN

γ > 0 x (x) hN (x) ≥ γ logN hN μγ (x) hN |γ| > 2 – √ = 0 μγ (x) ≥ ( + δ)logN hN 2 – √

Multiplicative chaos

Multiplicative chaos and -thick points Consider the set of -thick points This set contains points where is of the order of its variance rather than its standard deviation. It follows from the multiplicative chaos convergence that for any , in probability,

γ γ = {x ∈ [−1,1] : ± (x) ≥ ±γ logN}. T ±γ

N

hN (x) hN γ ∈ (− , ) ∖ {0} 2 – √ 2 – √ = − . lim

N→∞

log| | T γ

N

logN γ2 2

Multiplicative chaos

Freezing transition Another consequence of the multiplicative chaos convergence is that in probability. In the physics literature, this is called a freezing transition of the random energy landscape (cf. FYODOROV-BOUCHAUD '08, FYODOROV-LE DOUSSAL-RUSSO '12, FYODOROV-KEATING '14 for CUE).

log( dx) = { , lim

N→∞

1 logN ∫

1 −1

eγ

(x) hN

/2 γ2 γ − 1 2 – √

if γ ≤

2 – √

if γ ≥

2 – √ hN

Exponential moment estimates

Convergence to multiplicative chaos The key technical input to prove convergence of to consists of detailed asymptotic estimates as for exponential moments of the form These can also be written as Hankel determinants with Asymptotics are known (CHARLIER '18) for fixed and for independent of .

μγ

N

μ N → ∞ E . e

(x)+ (y)+ W( ) γ1hN γ2hN ∑N

j=1

λj

(x,y; , ;W) = det , DN γ1 γ2 ( f(λ;x,y; , ;W)dλ) ∫

R

λi+j γ1 γ2

N−1 i,j=0

f(λ;x,y; , ;W) = . γ1 γ2 e

π + π +W(λ)−NV(λ) 2 √ γ11{λ≤x} 2 √ γ21{λ≤y}

x ≠ y ∈ (−1,1) W N

Exponential moment estimates

Two merging singularities as , where the error term is uniform for , for sufficiently small. Method of proof We prove this using a method similar to one sed for Toeplitz determinants with merging Fisher-Hartwig singularities (C-KRASOVSKY '15) and Hankel determinants with merging root singularities (C-FAHS '16), based on a Riemann-Hilbert approach.

log ( , ; , ;0) = log ( ; + ;0) + π N d DN x1 x2 γ1 γ2 DN x1 γ1 γ2 2 – √ γ2 ∫

x2 x1

μV − max{0,log(| − |N)} + O(1), γ1γ2 x1 x2 N → ∞ −1 + δ < < < 1 − δ x1 x2 0 < − < δ x2 x1 δ

Exponential moment estimates

• dependent

Assume that is a sequence of functions which are analytic and uniformly bounded on a suitable domain which does not shrink too fast with . as , uniformly for in any fixed compact subset of , where

N W W = WN N log ( , ; , ; ) = log ( , ; , ;0) DN x1 x2 γ1 γ2 WN DN x1 x2 γ1 γ2 + N ∫ d + σ( + U ( ) + o(1), WN μV 1 2 WN)2 ∑

j=1 2

γj 2 – √ 1 − x2

j

− − − − − √ WN xj N → ∞ ( , ) x1 x2 (−1,1)2 σ(f = (x) (y) dxdy, (Uw)(x) = P.V. . )2 ∬

§2 f ′

f ′ Σ(x,y) 2π2 1 π ∫

1 −1

w(t) x − t dt 1 − t2 − − − − − √

Exponential moment estimates

Finally, we need also asymptotics for Hankel determinants with one singularity tending to the edge . This is needed for the upper bound estimate for the maximum of . Singularity close to the edge as , with the error term uniform for all , with sufficiently large.

±1 hN log = πγN d (ξ) + logN + log(1 − ) + O(1), (x;γ;0) DN (x;0;0) DN 2 – √ ∫

x −1

μV γ2 2 3γ2 4 x2 N → ∞ |x| ≤ 1 − MN −2/3 M

Overview

Summary of the method

• 1. Hankel determinant asymptotics

Convergence of to a multiplicative chaos measure Estimates for -thick points Estimates for the lower bound of

• 2. Hankel determinant asymptotics

Estimates for the upper bound of via one- moment method

• 3. Estimates for extrema of

Estimates for global rigidity of eigenvalues

⟹ dx

(x) eγhN Eeγ

(x) hN

μγ ⟹ γ ⟹ maxhN ⟹ maxhN hN ⟹

Simulations

Histogram of GUE eigenvalues for Normalized eigenvalue counting function for .

N = 300 hN N = 300

Simulations

with with with

eγ

(x) hN

Eeγ

(x) hN

γ = 0.5

eγ

(x) hN

Eeγ

(x) hN

γ = 1.0

eγ

(x) hN

Eeγ

(x) hN

γ = 1.4

Simulations

Histogram of GUE eigenvalues for Normalized eigenvalue counting function for .

N = 6000 hN N = 6000

Simulations

with with with

eγ

(x) hN

Eeγ

(x) hN

γ = 0.3

eγ

(x) hN

Eeγ

(x) hN

γ = 0.5

eγ

(x) hN

Eeγ

(x) hN

γ = 1.4

The end Thank you for your attention!