Smallest eigenvalue distributions for two classes of $β$-Jacobi ensembles
Provides theoretical results for eigenvalue distributions relevant to randomized algorithms in numerical linear algebra, but the contribution is incremental as it extends known results to new parameter ranges.
The paper derives exact and limiting smallest eigenvalue distributions for two classes of β-Jacobi ensembles not previously covered, using multivariate hypergeometric functions. Asymptotic results are obtained for special β values including β ∈ 2ℕ₊ and β=1.
We compute the exact and limiting smallest eigenvalue distributions for two classes of $β$-Jacobi ensembles not covered by previous studies. In the general $β$ case, these distributions are given by multivariate hypergeometric ${}_2F_{1}^{2/β}$ functions, whose behavior can be analyzed asymptotically for special values of $β$ which include $β\in 2\mathbb{N}_{+}$ as well as for $β= 1$. Interest in these objects stems from their connections (in the $β= 1,2$ 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}.