A convergence framework for Airy$_\beta$ line ensemble via pole evolution

TL;DR

This paper introduces a new convergence framework for the Airy$_\beta$ line ensemble using pole evolution of stochastic meromorphic functions, establishing its universality as the edge limit in various random matrix models.

Contribution

It develops a novel pole evolution approach to prove convergence to the Airy$_\beta$ line ensemble, extending universality results to multiple continuous-time processes.

Findings

Proves convergence of Dyson Brownian motions to Airy$_\beta$ ensemble.

Establishes universality for Laguerre and Jacobi processes.

Provides a unified framework for edge limit proofs in random matrix theory.

Abstract

The Airy$_\beta$ line ensemble is an infinite sequence of random curves. It is a natural extension of the Tracy-Widom$_\beta$ distributions, and is expected to be the universal edge scaling limit of a range of models in random matrix theory and statistical mechanics. In this work, we provide a framework of proving convergence to the Airy$_\beta$ line ensemble, via a characterization through the pole evolution of meromorphic functions satisfying certain stochastic differential equations. Our framework is then applied to prove the universality of the Airy$_\beta$ line ensemble as the edge limit of various continuous time processes, including Dyson Brownian motions with general $\beta$ and potentials, Laguerre processes and Jacobi processes.