Large Deviations Asymptotics of Rectangular Spherical Integral

TL;DR

This paper investigates the Dyson Bessel process related to rectangular matrix Brownian motions, establishing a large deviation principle for its empirical density, and derives asymptotics for rectangular spherical integrals as matrix dimensions grow.

Contribution

It introduces a large deviation principle for the Dyson Bessel process and derives asymptotic formulas for rectangular spherical integrals in the large dimension limit.

Findings

Established a large deviation principle for the Dyson Bessel process.

Derived asymptotic formulas for rectangular spherical integrals as matrix sizes tend to infinity.

Provided insights into the behavior of singular values of rectangular matrix Brownian motions.

Abstract

In this article we study the Dyson Bessel process, which describes the evolution of singular values of rectangular matrix Brownian motions, and prove a large deviation principle for its empirical particle density. We then use it to obtain the asymptotics of the so-called rectangular spherical integrals as $m,n$ go to infinity while $m/n$ converges.