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 F-2(1)2/beta functions, whose behavior can be analyzed asymptotically for special values of beta which include beta is an element of 2N(+) 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 [J. Demmel, I. Dumitriu, and O. Holtz, "Fast linear algebra is stable," Numer. Math. 108, 59-91 (2007); G. Ballard, J. Demmel, and I. Dumitriu, "Minimizing communication for eigenproblems and the singular value decomposition," 2010. Preprint]. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4748969]