Extreme eigenvalue distributions of Jacobi ensembles: new exact representations, asymptotics and finite size corrections

Abstract

Let W1 and W2 be independent complex central Wishart matrices with m1 and m2 degrees of freedom respectively. This paper is concerned with the extreme eigenvalue distributions of double-Wishart matrices (W1+W2)^-1 W1, which are analogous to those of F matrices W1 W2^-1 and those of the Jacobi unitary ensemble (JUE). Defining alpha1=m1-n and alpha2=m2-n with m1,m2>n, we derive new exact distribution formulas in terms of (alpha1+alpha2)-dimensional matrix determinants, with entries involving derivatives of Legendre polynomials. This provides a convenient exact representation, while facilitating a direct large-n analysis with alpha1 and alpha2 fixed (i.e., under the so-called “hard-edge” scaling limit). The analysis is based on new asymptotic properties of Legendre polynomials and their relation with Bessel functions that are here established. Specifically, we present limiting formulas for the smallest and largest eigenvalue distributions as n -> infinity in terms of alpha1- and alpha2-dimensional determinants respectively, which agrees with expectations from known universality results involving the JUE and the Laguerre unitary ensemble (LUE). We also derive finite-n corrections for the asymptotic extreme eigenvalue distributions under hard-edge scaling, giving new insights on universality by comparing with corresponding correction terms derived recently for the LUE. Our derivations are based on elementary algebraic manipulations and properties of Legendre polynomials, differing from existing results on double-Wishart and related models which often involve Fredholm determinants, Painlevé differential equations, or hypergeometric functions of matrix arguments.