Extreme gaps between eigenvalues of Wigner matrices

Paul Bourgade

doi:10.4171/jems/1141

Abstract

This paper proves universality of the distribution of the smallest and largest gaps between eigenvalues of generalized Wigner matrices, under some smoothness assumption for the density of the entries. The proof relies on the Erd{\H o}s-Schlein-Yau dynamic approach. We exhibit a new observable that satisfies a stochastic advection equation and reduces local relaxation of the Dyson Brownian motion to a maximum principle. This observable also provides a simple and unified proof of universality in the bulk and at the edge, which is quantitative. To illustrate this, we give the first explicit rate of convergence to the Tracy-Widom distribution for generalized Wigner matrices.

Highlights

1.1 Extreme statistics in random matrix theory
With a different method Soshnikov [49] computed the distribution of the smallest gap for general translation invariant determinantal point processes in large boxes: properly rescaled the smallest gap converges, with the same limiting distribution function e−x3 . √Vinson gave heuristics suggesting that the largest gap between eigenvalues in the bulk should be of order log N /N, with Poissonian fluctuations around this limit, a problem popularized by Diaconis [16]
Feng and Wei investigated the smallest gaps beyond the determinantal case, characterizing their asymptotics for the circular β ensembles [27]

Summary

Introduction

1.1 Extreme statistics in random matrix theory. The study of extreme spacings in random spectra was initially limited to integrable models. Integrating (1.16) over ν ∈ (0, 1), we obtain (1.6), which is the main estimate for theorems 1.2 and 1.4 To summarize this proof sketch, the observable (1.11) and the stochastic advection equation (1.12) it satisfies are new ingredients to quantify relaxation of the Dyson Brownian motion and obtain universality beyond microscopic scales. [2, 26, 46]) have written the stochastic advection equation for the resolvent of a matrix following the Dyson Brownian motion dynamics These resolvent dynamics can be used for regularization and universality purpose, as proved first in [39], for eigenvalues statistics at the edge of deformed Wigner matrices. The Stieltjes transform is a specialization of our observable ft when uk (t)

Stochastic advection equation

Relaxation from a maximum principle

Extreme gaps

Rate of convergence to Tracy-Widom