Continuum limits of random matrices and the Brownian carousel

TL;DR

This paper demonstrates that eigenvalues of Gaussian ensembles converge to a translation invariant process described by the Brownian carousel, revealing new insights into their continuum limits and phase transitions.

Contribution

It introduces the Brownian carousel as a geometric framework for analyzing continuum limits of random matrices and characterizes the properties of the Sine_beta process.

Findings

Eigenvalues converge to Sine_beta process away from spectral edges.

Gap probability is continuous in gap size and beta.

Identifies a phase transition at beta=2 in the SDE version of the Brownian carousel.

Abstract

We show that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general beta siblings converge to Sine_beta, a translation invariant point process. This process has a geometric description in term of the Brownian carousel, a deterministic function of Brownian motion in the hyperbolic plane. The Brownian carousel, a description of the a continuum limit of random matrices, provides a convenient way to analyze the limiting point processes. We show that the gap probability of Sine_beta is continuous in the gap size and $\beta$, and compute its asymptotics for large gaps. Moreover, the stochastic differential equation version of the Brownian carousel exhibits a phase transition at beta=2.