Airy$_\beta$ line ensemble and its Laplace transform

TL;DR

This paper introduces the Airy$_\beta$ line ensemble as a universal limit object in random matrix theory and statistical mechanics, characterizing it via Laplace transforms and proving its emergence as a limit for eigenvalues in Dyson Brownian Motion and G$\beta$E corners process.

Contribution

The paper defines and characterizes the Airy$_\beta$ line ensemble using integral formulas for joint moments and establishes its universality as a limit in key stochastic models.

Findings

Trajectories of largest eigenvalues in Dyson Brownian Motion converge to the Airy$_\beta$ line ensemble.

Extreme particles in G$\beta$E corners process converge to the Airy$_\beta$ line ensemble.

Integral formulas for multi-time moments are derived for the ensemble.

Abstract

The Airy$_\beta$ line ensemble is a random collection of continuous curves, which should serve as a universal edge scaling limit in problems related to eigenvalues of random matrices and models of 2d statistical mechanics. This line ensemble unifies many existing universal objects including Tracy-Widom distributions, eigenvalues of the Stochastic Airy Operator, Airy$_2$ process from the KPZ theory. Here $\beta>0$ is a real parameter governing the strength of the repulsion between the curves. We introduce and characterize the Airy$_\beta$ line ensemble in terms of the Laplace transform, by producing integral formulas for its joint multi-time moments. We prove two asymptotic theorems for each $\beta>0$: the trajectories of the largest eigenvalues in the Dyson Brownian Motion converge to the Airy$_\beta$ line ensemble; the extreme particles in the G$\beta$E corners process converge to the same limit. The proofs are based on the convergence of random walk expansions for the multi-time moments of prelimit objects towards their Brownian counterparts. The expansions are produced through Dunkl differential-difference operators acting on multivariate Bessel generating functions.