Smallest eigenvalue distributions for two classes of $\beta$-Jacobi ensembles

TL;DR

This paper derives exact and asymptotic distributions for the smallest eigenvalues in two classes of beta-Jacobi ensembles, extending previous results and connecting to random matrix theory and algorithms.

Contribution

It provides the first explicit formulas for these distributions in the general beta case, including asymptotic analysis for special beta values.

Findings

Distributions expressed via multivariate hypergeometric functions.

Asymptotic behavior analyzed for beta in 2N+ and beta=1.

Connections established to principal submatrices of Haar matrices.

Abstract

We compute the exact and limiting smallest eigenvalue distributions for two classes of $\beta$-Jacobi ensembles not covered by previous studies. In the general $\beta$ case, these distributions are given by multivariate hypergeometric ${}_2F_{1}^{2/\beta}$ functions, whose behavior can be analyzed asymptotically for special values of $\beta$ which include $\beta \in 2\mathbb{N}_{+}$ as well as for $\beta = 1$. Interest in these objects stems from their connections (in the $\beta = 1,2$ cases) to principal submatrices of Haar-distributed (orthogonal, unitary) matrices appearing in randomized, communication-optimal, fast, and stable algorithms for eigenvalue computations \cite{DDH07}, \cite{BDD10}.