The beta-Wishart ensemble

Abstract

We introduce a “broken-arrow” matrix model for the β-Wishart ensemble, which improves on the traditional bidiagonal model by generalizing to non-identity covariance parameters. We prove that its joint eigenvalue density involves the correct hypergeometric function of two matrix arguments, and a continuous parameter β > 0. If we choose β = 1, 2, 4, we recover the classical Wishart ensembles of general covariance over the reals, complexes, and quaternions. Jack polynomials are often defined as the eigenfunctions of the Laplace-Beltrami operator. We prove that Jack polynomials are in addition eigenfunctions of an integral operator defined as an average over a β-dependent measure on the sphere. When combined with an identity due to Stanley, we derive a definition of Jack polynomials. An efficient numerical algorithm is also presented for simulations. The algorithm makes use of secular equation software for broken arrow matrices currently unavailable in the popular technical computing languages. The simulations are matched against the cdfs for the extreme eigenvalues. The techniques here suggest that arrow and broken arrow matrices can play an important role in theoretical and computational random matrix theory including the study of corners processes. We provide a number of simulations illustrating the extreme eigenvalue distributions that are likely to be useful for applications. We also compare the n → ∞ answer for all β with the free-probability prediction.