Center of mass distribution of the Jacobi unitary ensembles: Painleve V, asymptotic expansions

TL;DR

This paper analyzes the probability distribution of the center of mass in Jacobi unitary ensembles, deriving its exponential moment generating function and exploring connections to Painleve V and asymptotic expansions.

Contribution

It introduces a novel computation of the exponential moment generating function for the center of mass in Jacobi ensembles, linking it to Painleve V and asymptotic analysis.

Findings

Explicit form of the exponential moment generating function

Connection to Painleve V transcendents

Asymptotic expansions for large matrix size

Abstract

In this paper, we study the probability density function, $\mathbb{P}(c,\alpha,\beta, n)\,dc$, of the center of mass of the finite $n$ Jacobi unitary ensembles with parameters $\alpha\,>-1$ and $\beta >-1$; that is the probability that ${\rm tr}M_n\in(c, c+dc),$ where $M_n$ are $n\times n$ matrices drawn from the unitary Jacobi ensembles. We first compute the exponential moment generating function of the linear statistics $\sum_{j=1}^{n}\,f(x_j):=\sum_{j=1}^{n}x_j,$ denoted by $\mathcal{M}_f(\lambda,\alpha,\beta,n)$.