Gap Probability Distribution of the Jacobi Unitary Ensemble: An Elementary Treatment, from Finite $n$ to Double Scaling

TL;DR

This paper provides an elementary approach to analyze the gap probability in the Jacobi unitary ensemble, deriving differential equations and connecting the problem to Painlevé V through double scaling limits.

Contribution

It introduces a new elementary method using ladder operators to derive differential equations for gap probabilities in the Jacobi ensemble, linking to Painlevé V.

Findings

Derived second order differential equations for key quantities

Connected the large $n$ limit to Painlevé V equation

Reduced complex differential equations to a known integrable form

Abstract

In this paper, we study the gap probability problem of the (symmetric) Jacobi unitary ensemble of Hermitian random matrices, namely the probability that the interval $(-a,a)\:(0<a<1)$ is free of eigenvalues. Using the ladder operator technique for orthogonal polynomials and the associated supplementary conditions, we derive three quantities instrumental in the gap probability, denoted by $H_{n}(a)$, $R_{n}(a)$ and $r_{n}(a)$. We find that each one satisfies a second order differential equation. We show that after a double scaling, the large second order differential equation in the variable $a$ with $n$ as parameter satisfied by $H_{n}(a)$, can be reduced to the Jimbo-Miwa-Okamoto $\sigma$ form of the Painlev\'{e} V equation.