Asymptotic Gap Probability Distributions of the Gaussian Unitary Ensembles and Jacobi Unitary Ensembles

TL;DR

This paper investigates the asymptotic behavior of gap probabilities in Gaussian and Jacobi unitary ensembles, linking them to eigenvalue distributions and special functions, and provides explicit asymptotic expansions and constants.

Contribution

It introduces a novel approach to compute gap probabilities using eigenvalue distributions and derives asymptotic expansions with explicit constants for GUE and JUE.

Findings

Gap probabilities expressed as products of eigenvalue distributions.

Derived asymptotic expansions for smallest eigenvalue distributions.

Explicit constants in asymptotics involving Barnes G-function.

Abstract

In this paper, we address a class of problems in unitary ensembles. Specifically, we study the probability that a gap symmetric about 0, i.e. $(-a,a)$ is found in the Gaussian unitary ensembles (GUE) and the Jacobi unitary ensembles (JUE) (where in the JUE, we take the parameters $\alpha=\beta$). By exploiting the even parity of the weight, a doubling of the interval to $(a^2,\infty)$ for the GUE, and $(a^2,1)$, for the (symmetric) JUE, shows that the gap probabilities maybe determined as the product of the smallest eigenvalue distributions of the LUE with parameter $\alpha=-1/2,$ and $\alpha=1/2$ and the (shifted) JUE with weights $x^{1/2}(1-x)^{\beta}$ and $x^{-1/2}(1-x)^{\beta}$ The $\sigma$ function, namely, the derivative of the log of the smallest eigenvalue distributions of the finite-$n$ LUE or the JUE, satisfies the Jimbo-Miwa-Okamoto $\sigma$ form of $P_{V}$ and $P_{VI}$, although in the shift Jacobi case, with the weight $x^{\alpha}(1-x)^{\beta},$ the $\beta$ parameter does not show up in the equation. We also obtain the asymptotic expansions for the smallest eigenvalue distributions of the Laguerre unitary and Jacobi unitary ensembles after appropriate double scalings, and obtained the constants in the asymptotic expansion of the gap probablities, expressed in term of the Barnes $G-$ function valuated at special point.