Abstract
Let W1 and W2 be independent n×n 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)−1W1, which are analogous to those of F matrices W1W2−1 and those of the Jacobi unitary ensemble (JUE). Defining α1=m1−n and α2=m2−n with m1, m2≥n, we derive new exact distribution formulas in terms of (α1+α2)-dimensional matrix determinants, with entries involving derivatives of Legendre polynomials. This provides a convenient exact representation, while facilitating a direct large-n analysis with α1 and α2 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→∞ in terms of α1- and α2-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.
Highlights
Double Wishart random matrices, defined as W = (W1 + W2)−1W1, with W1 and W2 n × n Wishart with m1 and m2 degrees of freedom respectively, are an important class of random matrix models
A goal of this paper is to provide finite-n corrections for the extreme eigenvalue distributions of W, and for the F model and the classical Jacobi unitary ensemble (JUE), under hard-edge asymptotics
We unveil a striking connection of these exact distributions, classically associated with the Jacobi polynomials, with the simpler Legendre polynomials. This new connection 65 allows us to firstly give an explicit proof that shows that the extreme eigenvalue distributions of W can be expressed in terms of α1- and α2-dimensional determinants involving Bessel functions, without resorting to study correlation kernels
Summary
We unveil a striking connection of these exact distributions, classically associated with the Jacobi polynomials, with the simpler Legendre polynomials This new connection 65 allows us to firstly give an explicit proof that shows that the extreme eigenvalue distributions of W can be expressed in terms of α1- and α2-dimensional determinants involving Bessel functions, without resorting to study correlation kernels. Following similar 70 manipulations, we provide finite-n corrections for the extreme eigenvalue distributions of W, giving insights on the universality for the JUE and LUE at the left edge of the spectrum support To this end, we derive new asymptotic results for the Legendre and associated Legendre polynomials, which are non-standard and may be of independent interest
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
