Gap Probabilities for Edge Intervals in Finite Gaussian and Jacobi Unitary Matrix Ensembles

TL;DR

This paper derives exact formulas and differential equations for gap probabilities in finite Gaussian and Jacobi unitary ensembles, analyzing their asymptotics and connections to Painlevé transcendents.

Contribution

It introduces new differential equations for gap probabilities in finite ensembles and links these to Painlevé equations, providing explicit solutions for small N.

Findings

Explicit formulas for N=1,2 gap probabilities

Asymptotic expansions for large gaps

Connections to Painlevé-V and Painlevé-VI equations

Abstract

The probabilities for gaps in the eigenvalue spectrum of the finite dimension $ N \times N $ random matrix Hermite and Jacobi unitary ensembles on some single and disconnected double intervals are found. These are cases where a reflection symmetry exists and the probability factors into two other related probabilities, defined on single intervals. Our investigation uses the system of partial differential equations arising from the Fredholm determinant expression for the gap probability and the differential-recurrence equations satisfied by Hermite and Jacobi orthogonal polynomials. In our study we find second and third order nonlinear ordinary differential equations defining the probabilities in the general $N$ case. For N=1 and N=2 the probabilities and thus the solution of the equations are given explicitly. An asymptotic expansion for large gap size is obtained from the equation in the Hermite case, and also studied is the scaling at the edge of the Hermite spectrum as $ N \to \infty $, and the Jacobi to Hermite limit; these last two studies make correspondence to other cases reported here or known previously. Moreover, the differential equation arising in the Hermite ensemble is solved in terms of an explicit rational function of a {Painlev\'e-V} transcendent and its derivative, and an analogous solution is provided in the two Jacobi cases but this time involving a {Painlev\'e-VI} transcendent.