Computable structural formulas for the distribution of the $\beta$-Jacobi edge eigenvalues

TL;DR

This paper derives explicit formulas for the distribution of edge eigenvalues in the beta-Jacobi ensemble, enabling precise computation of their probabilities using a recursive scheme based on differential-difference systems.

Contribution

It introduces a finite-sum representation for the gap probabilities at the spectrum edge and develops a recursive method to compute these probabilities for the beta-Jacobi ensemble.

Findings

Finite-sum formulas for edge eigenvalue distributions in the beta-Jacobi ensemble.

A recursive computational scheme based on differential-difference systems.

Applicable to various parameter ranges of the ensemble.

Abstract

The Jacobi ensemble is one of the classical ensembles of random matrix theory. Prominent in applications are properties of the eigenvalues at the spectrum edge, specifically the distribution of the largest (e.g. Roy's largest root test in multivariate statistics) and smallest (e.g.~condition numbers of linear systems) eigenvalues. We identify three ranges of parameter values for which the gap probability determining these distributions is a finite sum with respect to particular bases, and moreover make use of a certain differential-difference system fundamental in the theory of the Selberg integral to provide a recursive scheme to compute the corresponding coefficients.