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 ✓ Global statistics of eigenvalues: limiting eigenvalue distribution, macroscopic linear statistics ... ✓ Local statistics of eigenvalues: universal local correlations, extreme eigenvalue distribution ✓ In this talk: maximal fluctuation of eigenvalues around their classical positions

Global rigidity in the GUE Classical GUE eigenvalue locations λ 1 ≤ λ 2 ≤ ⋯ ≤ λ N M Let be the eigenvalues of a GUE matrix of size N × N , normalized such that the eigenvalue distribution converges to a [−1,1] semi-circle law on . M M i , j (Equivalently, is Hermitian and the independent entries are iid (real 1 on the diagonal, complex otherwise) Gaussians with variance .) 4 n − − − − − π ∫ κ j j 2 1 − x 2 √ Classical locations ,…, ∈ [−1,1] are given by dx = . κ 1 κ N −1 N

Global rigidity in the GUE Global rigidity What can we say for large N about the distribution of the normalized maximal fluctuation of eigenvalues − − − − − 2 1 − κ 2 M N := max { | λ j − κ j | } ? √ j j =1,…, N π

Global rigidity in the GUE Upper bound for generalized Wigner matrices (E RDOS -Y AU -Y IN '12 ) (log N ) α log log N α ′ log log N P ( M N ≥ ) ≤ C exp ( − c (log N ) ) N Lower bound for GUE (G USTAVSSON '05) − − − − − N – 1 − κ 2 2 √ 2 ( λ j − κ j ) → N (0,1) √ − − − − − j √ log N for δ ≤ j ≤ (1 − δ ) N , which implies (non-optimal) lower bounds for M N .

Global rigidity in the GUE Theorem (C-Fahs-Lambert-Webb '18) For any ϵ > 0 , we have log N log N lim P ( (1 − ϵ ) < M N < (1 + ϵ ) ) = 1. N →∞ πN πN Unitary invariant ensembles A similar result holds for unitary invariant ensembles with eigenvalue distribution 1 2 ∏ e − NV ( λ j ) ∣ λ j ∣ ∏ ∣ λ i − d λ j ∣ Z N 1≤ i < j ≤ N 1≤ j ≤ N V for real analytic with sufficient growth at ±∞ .

Global rigidity in unitary invariant ensembles Equilibrium measure and classical locations Semi-circle law is then replaced by the equilibrium measure μ V minimizing | −1 ∫ log| x − y dμ ( x ) dμ ( y ) + ∫ V ( x ) dμ ( x ). R × R R We assume that μ V is one-cut regular, and that the support is [−1,1] for convenience. The classical locations κ 1 ,…, κ N ∈ [−1,1] are now defined by j ∫ κ j −1 μ V d ( x ) = . N

Global rigidity in unitary invariant ensembles Theorem (C-Fahs-Lambert-Webb '18) For any ϵ > 0 , we have (1 − ϵ )log N (1 + ϵ )log N dμ V lim P ( < max { ( κ j λ j )| − κ j | } < ) = 1. N →∞ πN dx πN

Global rigidity in unitary invariant ensembles Eigenvalue counting function We prove this via the extrema of the normalized eigenvalue counting function x – h N ( x ) = √ 2 π ( ∑ 1 − N ∫ d μ V ) , x ∈ R . ≤ x λ j −1 1≤ j ≤ N δ > 0 Namely, we prove that for any , – – lim P [ (1 − δ ) √ 2 log N ≤ max { ± h N ( x ) } ≤ (1 + δ ) √ 2 log N ] = 1. N →∞ x ∈ R dμ V ∫ κ j Heuristically, we expect h N λ j ( ) = d μ V ( x ) ≈ dx κ j ( )( κ j − λ j ), which λ j explains the connection between global rigidity and the maximum of the normalized eigenvalue counting function.

Extreme values of the eigenvalue counting function Extreme of log-correlated fields h N N behaves for large like a stochastic process with log-correlations (J OHANSSON '98) How to estimate extrema of log-correlated processes? This question has been studied in different contexts. ζ ✓ Riemann function and CUE (F YODOROV -H IARY -K EATING '12, A RGUIN -B ELIUS -B OURGADE '16, C HHAIBI -M ADAULE -N AJNUDEL '16) ✓ Circular Beta Ensemble and Sine Beta process (C HHAIBI - M ADAULE -N AJNUDEL '16, P AQUETTE -Z EITOUNI '16, H OLCOMB - P AQUETTE '18) ✓ Characteristic polynomial in unitary invariant ensembles (F YODOROV -S IMM '14, L AMBERT -P AQUETTE '18)

Extreme values of the eigenvalue counting function Multiplicative chaos Powerful tools to study such extrema come from the theory of multiplicative chaos ✓ General theory (K AHANE '85, R HODES -V ARGAS '14, B ERESTYCKI '15) ✓ Applied to Circular Unitary Ensemble (F YODOROV -K EATING '14, W EBB '15, B ERESTYCKI -W EBB -W ONG '18, L AMBERT -O STROVSKY - S IMM '18) Exponential moments Crucial input for this method: good control of exponential moments E e γ h N ( x ) γ 1 h N x 1 ( )+ γ 2 h N x 2 ( ) and E e for large N

Extreme values of the eigenvalue counting function Upper bound estimates { ± ( x ) } Upper bound for max x ∈ I h N can be obtained using an elementary one-moment method. 1. max { ± h N ( x ) } ≤ max { ± h N κ j ( ) } + 1. x ∈ I j : κ j ∈ I 2. By a union bound and Markov's inequality, E e γ h N κ j ( ) P ( max { h N κ j ( ) } > Y ) ≤ ∑ P ( h N κ j ( ) > Y ) ≤ ∑ . e γY j : κ j ∈ I j : κ j ∈ I j : κ j ∈ I E e γ h N ( x ) 3. Substitute large N asymptotics for and choose Y as big as possible such that rhs decays for some . γ

Extreme values of the eigenvalue counting function Upper bound estimates E e γ ( x ) e − NV ( λ ) e γ 1 λ ≤ x h N is a Hankel determinant with discontinuous weight , and large N asymptotics for such Hankel determinants are known for x ∈ (−1 + δ ,1 − δ ) (I TS -K RASOVSKY '08 for GUE, C HARLIER '18 for one-cut regular unitary invariant ensembles): γ 2 e γ h N ( x ) 2 E ≤ , x ∈ (−1 + δ ,1 − δ ). C γ N x To extend this to all eigenvalues, we need a similar result for close to ±1 . We prove γ 2 3 γ 2 e γ h N ( x ) x 2 ) N −2/3 C ′ 2 4 E ≤ γ N (1 − , | x | ≤ 1 − m .

Extreme values of the eigenvalue counting function Lower bound estimates Optimal lower bound estimates are much harder to obtain, and require to h N investigate the log-correlated structure of . Log-correlated structure h N behaves for large N (J OHANSSON '98) like a Gaussian process X ( x ) with logarithmic covariance kernel − − − − − − − − − − 1 − x 2 1 − y 2 ∣ ∣ √ ∣ 1 − xy + √ Σ( x , y ) := log ∣ . x − y ∣ ∣

Multiplicative chaos Maximum of the eigenvalue counting function For studying the maximum of h N , we prove that the random measure e γ h N ( x ) μ γ d = dx , γ ∈ R N E e γ h N ( x ) converges weakly in distribution to a multiplicative chaos measure which can be formally written as (cf. K AHANE '85, R HODES -V ARGAS '10, B ERESTYCKI '17, B ERESTYCKI -W EBB -W ONG '17) e γX ( x ) μ γ d ( x ) = dx . E e γX ( x )

Multiplicative chaos Extreme values μ γ It will turn out that the extreme values of the limiting measure will lead h N us to estimates for extreme values of . Heuristics e γ hN ( x ) μ γ Heuristically, the random measure d ( x ) = dx is expected to be N E e γ hN ( x ) dominated for γ > 0 by -values where x h N ( x ) is exceptionally large, namely h N ( x ) ≥ γ log N and it is natural to expect that the multiplicative μ γ chaos measure will give us information about large values of h N ( x ) . – μ γ √ For | γ | > 2 , = 0 , which suggests heuristically that values where – √ h N ( x ) ≥ ( 2 + δ )log N are unlikely to occur.

Multiplicative chaos Multiplicative chaos and -thick points γ Consider the set of -thick points γ T ± γ = { x ∈ [−1,1] : ± h N ( x ) ≥ ± γ log N }. N This set contains points where h N ( x ) is of the order of its variance rather than its standard deviation. It follows from the multiplicative chaos – – √ √ convergence that for any γ ∈ (− 2 , 2 ) ∖ {0} , in probability, T γ γ 2 log| | N = − . lim N →∞ log N 2

Multiplicative chaos Freezing transition Another consequence of the multiplicative chaos convergence is that – 1 γ 2 1 √ /2 if γ ≤ 2 e γ ( x ) h N lim log ( ∫ dx ) = { , – – N →∞ log N √ √ γ − 1 if γ ≥ 2 2 −1 in probability. In the physics literature, this is called a freezing transition of the random energy landscape h N (cf. F YODOROV -B OUCHAUD '08, F YODOROV -L E D OUSSAL -R USSO '12, F YODOROV -K EATING '14 for CUE).

Exponential moment estimates Convergence to multiplicative chaos μ γ The key technical input to prove convergence of to consists of μ N detailed asymptotic estimates as N → ∞ for exponential moments of the form ∑ N γ 1 h N ( x )+ γ 2 h N ( y )+ W ( λ j ) j =1 E . e These can also be written as Hankel determinants N −1 λ i + j ( x , y ; , ; W ) = det , D N γ 1 γ 2 ( ∫ f ( λ ; x , y ; γ 1 γ 2 , ; W ) dλ ) R i , j =0 √ 2 π γ 1 1 { λ ≤ x } + √ 2 π γ 2 1 { λ ≤ y } + W ( λ )− NV ( λ ) with f ( λ ; x , y ; γ 1 γ 2 , ; W ) = e . x ≠ y ∈ (−1,1) W Asymptotics are known (C HARLIER '18) for fixed and for N independent of .