The hard-to-soft edge transition: exponential moments, central limit theorems and rigidity

TL;DR

This paper studies the transition of eigenvalue statistics from a hard edge to a soft edge in large random matrices, providing detailed asymptotics, limit theorems, and rigidity bounds for the associated Bessel process.

Contribution

It offers new precise asymptotics, central limit theorems, and rigidity bounds for eigenvalue distributions near the hard-to-soft edge transition in random matrices.

Findings

Exponential moment asymptotics including the constant term

Asymptotics for expectation and variance of the counting function

Several central limit theorems and a global rigidity upper bound

Abstract

The local eigenvalue statistics of large random matrices near a hard edge transitioning into a soft edge are described by the Bessel process associated with a large parameter $\alpha$. For this point process, we obtain 1) exponential moment asymptotics, up to and including the constant term, 2) asymptotics for the expectation and variance of the counting function, 3) several central limit theorems and 4) a global rigidity upper bound.