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

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 Painlevé V equation.

arXiv comment20 pages

References in corpus (5)

• Fredholm Determinants, Differential Equations and Matrix Models
• Painlevé VI and Hankel determinants for the generalized Jacobi Weight
• Gap Probabilities for Edge Intervals in Finite Gaussian and Jacobi Unitary Matrix Ensembles
• Continuous and discrete Painleve equations arising from the gap probability distribution of the finite $n$ Gaussian Unitary Ensembles
• PDEs satisfied by extreme eigenvalues distributions of GUE and LUE

Cited by in corpus (10)

• Asymptotic Gap Probability Distributions of the Gaussian Unitary Ensembles and Jacobi Unitary Ensembles
• Painlevé Transcendents and the Hankel Determinants Generated by a Discontinuous Gaussian Weight
• Orthogonal Polynomials, Asymptotics and Heun Equations
• Painlevé III${}^{\prime }$ and the Hankel Determinant Generated by a Singularly Perturbed Gaussian Weight
• Differential, Difference and Asymptotic Relations for Pollaczek-Jacobi Type Orthogonal Polynomials and Their Hankel Determinants
• Painlevé V, Painlevé XXXIV and the Degenerate Laguerre Unitary Ensemble
• Laguerre Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé V System
• Painlevé VI, Painlevé III and the Hankel Determinant Associated with a Degenerate Jacobi Unitary Ensemble
• Asymptotics of the Largest Eigenvalue Distribution of the Laguerre Unitary Ensemble
• On determinant expansions for Hankel operators