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. C 2013 AIP Publishing LLC. [http://dx.

Hypergeometric functions of matrix arguments and linear statistics of multi-spiked Hermitian matrix models

“…whilst, in addition, [DEKV13,Lemma 7] implies that C (α) τ (X ) = C (α) τ (X) for any τ with ℓ(τ ) ≤ r, and C (α) τ (X) = 0 if ℓ(τ ) > r. Together, these results yield…”

The Correlated Jacobi and the Correlated Cauchy–Lorentz Ensembles

“…Thus, it is reasonable to apply other methods than the standard Jack or Zonal polynomial approach [2,34,35].…”

Eigenvalue distributions of beta-Wishart matrices

“…Introduction. Recently, the classical real (β = 1), complex (β = 2), and quaternion (β = 4) Wishart random matrix ensembles were generalized to any β > 0 by what is now called the Beta-Wishart ensemble [2,9]. In this paper we derive the explicit distributions for the extreme eigenvalues and the trace of this ensemble as series of Jack functions and, in particular, in terms of the hypergeometric function of matrix argument.…”

“…Recently, it is becoming increasingly apparent that the classical matrix ensembles (and perhaps all of random matrix theory) generalizes from the Dyson's three-fold way (β = 1, 2, 4) [5] to any β > 0 [6]. Recent examples include the Beta-Hermite [3], Beta-Laguerre [3], Beta-Jacobi [4,7,12,14], and Beta-Wishart ensembles [2,9].…”

“…q (a; b; X, I) Following [2], an m × m Beta-Wishart matrix with n (n ≥ m) degrees of freedom and covariance matrix Σ (denoted A ∼ W (β) m (n, Σ)) has joint eigenvalue density…”