# Extreme Eigenvalue Distributions of Jacobi Ensembles: New Exact Representations, Asymptotics and Finite Size Corrections

## TL;DR

This paper develops new exact formulas and asymptotic analyses for the extreme eigenvalue distributions of Jacobi ensembles, providing finite-size corrections and insights into universality, using elementary algebraic methods and properties of Legendre polynomials.

## Contribution

It introduces new exact determinant formulas and asymptotic results for eigenvalues of Jacobi ensembles, with finite-size corrections, using elementary methods and Legendre polynomial properties.

## Key findings

- Derived exact distribution formulas involving matrix determinants and Legendre polynomials.
- Established large-$n$ asymptotics for extreme eigenvalues under hard-edge scaling.
- Provided finite-$n$ corrections that enhance understanding of universality in eigenvalue distributions.

## Abstract

Let $\mathbf{W}_1$ and $\mathbf{W}_2$ be independent $n\times n$ complex central Wishart matrices with $m_1$ and $m_2$ degrees of freedom respectively. This paper is concerned with the extreme eigenvalue distributions of double-Wishart matrices $(\mathbf{W}_1+\mathbf{W}_2)^{-1}\mathbf{W}_1$, which are analogous to those of F matrices ${\bf W}_1 {\bf W}_2^{-1}$ and those of the Jacobi unitary ensemble (JUE). Defining $\alpha_1=m_1-n$ and $\alpha_2=m_2-n$, we derive new exact distribution formulas in terms of $(\alpha_1+\alpha_2)$-dimensional matrix determinants, with elements involving derivatives of Legendre polynomials. This provides a convenient exact representation, while facilitating a direct large-$n$ analysis with $\alpha_1$ and $\alpha_2$ 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 $n \to \infty$ in terms of $\alpha_1$- and $\alpha_2$-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, differing from existing results on double-Wishart and related models which often involve Fredholm determinants, Painlev\'e differential equations, or hypergeometric functions of matrix arguments.