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

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}.