Diffusion at the random matrix hard edge

TL;DR

This paper characterizes the distribution of the smallest eigenvalues at the hard edge of beta random matrix ensembles using diffusion processes, revealing a transition between soft and hard edge laws across all beta values.

Contribution

It introduces a diffusion-based framework for describing the hard edge eigenvalue distribution and establishes a transition between soft and hard edge laws for all beta.

Findings

Eigenvalue distributions are described by a random diffusion generator.

A Riccati transformation yields a second diffusion description in terms of hitting laws.

A transition between soft and hard edge laws is proven for all beta.

Abstract

We show that the limiting minimal eigenvalue distributions for a natural generalization of Gaussian sample-covariance structures (the "beta ensembles") are described by the spectrum of a random diffusion generator. By a Riccati transformation, we obtain a second diffusion description of the limiting eigenvalues in terms of hitting laws. This picture pertains to the so-called hard edge of random matrix theory and sits in complement to the recent work of the authors and B. Virag on the general beta random matrix soft edge. In fact, the diffusion descriptions found on both sides are used here to prove there exists a transition between the soft and hard edge laws at all values of beta.