# Extreme eigenvalues of random matrices from Jacobi ensembles

## TL;DR

This paper derives two-term asymptotic formulas for the distribution of the smallest and largest eigenvalues in Jacobi beta-ensembles, revealing explicit expressions and correction terms for large matrices.

## Contribution

It provides new two-term asymptotic formulas for eigenvalue distributions in Jacobi beta-ensembles with explicit correction terms and special case formulas involving familiar functions.

## Key findings

- Explicit two-term asymptotic formulas derived
- First-order corrections proportional to distribution derivatives
- Special cases with explicit formulas involving Bessel functions

## Abstract

Two-term asymptotic formulae for the probability distribution functions for the smallest eigenvalue of the Jacobi $ \beta $-Ensembles are derived for matrices of large size in the r\'egime where $ \beta > 0 $ is arbitrary and one of the model parameters $ \alpha_1 $ is an integer. By a straightforward transformation this leads to corresponding results for the distribution of the largest eigenvalue. The explicit expressions are given in terms of multi-variable hypergeometric functions, and it is found that the first-order corrections are proportional to the derivative of the leading order limiting distribution function. In some special cases $ \beta = 2 $ and/or small values of $ \alpha_1 $, explicit formulae involving more familiar functions, such as the modified Bessel function of the first kind, are presented.