Feynman-Kac formula for the stochastic Bessel operator

TL;DR

This paper develops a stochastic process and functional framework to analyze the stochastic Bessel operator, advancing understanding of eigenvalue distributions at the hard edge of certain random matrix ensembles.

Contribution

It introduces a Feynman-Kac type representation for the stochastic Bessel semigroup and demonstrates convergence of associated functionals, extending the stochastic analysis at the hard edge.

Findings

Representation of matrix powers via Feynman-Kac formulas

Convergence of stochastic functionals to the Bessel semigroup

Insights into edge transition phenomena in random matrix models

Abstract

We introduce a stochastic process and functional that should describe the semigroup generated by the stochastic Bessel operator. Recently Gorin and Shkolnikov showed that the largest eigenvalues for certain random matrix ensembles with soft edge behavior can be understood by analyzing large powers of tridiagonal matrices, which converge to operators in the stochastic Airy semigroup. In this article we make some progress towards realizing Gorin and Shkolnikov's program at the random matrix hard edge. We analyze large powers of a suitable tridiagonal matrix model (a slight modification of the $\beta$-Laguerre ensemble). For finite $n$ we represent the matrix powers using Feynman-Kac type formulas, which identifies a sequence of stochastic processes $X^n$ and functionals $\Phi_n$. We show that $\Phi_n(X^n)$ converges in probability to the limiting functional $\Phi(X)$ for our proposed stochastic Bessel semigroup. We also discuss how the semigroup method may be used to understand transitions from a hard edge to a soft edge in the $\beta$-Laguerre models.