# Extreme gaps between eigenvalues of Wigner matrices

## TL;DR

This paper establishes the universality of the distribution of extreme eigenvalue gaps in generalized Wigner matrices, providing explicit convergence rates to Tracy-Widom distribution using a novel observable and the Dyson Brownian motion approach.

## Contribution

It introduces a new observable satisfying a stochastic advection equation, simplifying the proof of universality for eigenvalue gaps in Wigner matrices.

## Key findings

- Proves universality of eigenvalue gap distributions.
- Provides explicit convergence rates to Tracy-Widom distribution.
- Introduces a new observable for analyzing Dyson Brownian motion.

## 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.