On the ratio probability of the smallest eigenvalues in the Laguerre Unitary Ensemble

TL;DR

This paper investigates the probability distribution of the ratio of the second smallest to smallest eigenvalues in the Laguerre Unitary Ensemble, expressing it through Hankel determinants and Painlevé V equations, with asymptotic analysis.

Contribution

It derives a new integral representation for the ratio distribution and connects it to coupled Painlevé V equations, providing asymptotic behaviors in various regimes.

Findings

Distribution expressed via Hankel determinants with perturbed Laguerre weight

Limiting distribution involves functions satisfying coupled Painlevé V equations

Asymptotic behaviors analyzed for different regimes of parameters

Abstract

We study the probability distribution of the ratio between the second smallest and smallest eigenvalue in the $n\times n$ Laguerre Unitary Ensemble. The probability that this ratio is greater than $r>1$ is expressed in terms of an $n \times n$ Hankel determinant with a perturbed Laguerre weight. The limiting probability distribution for the ratio as $n\to\infty$ is found as an integral over $(0,\infty)$ containing two functions $q_{1}(x)$ and $q_{2}(x)$. These functions satisfy a system of two coupled Painlev\'e V equations, which are derived from a Lax pair of a Riemann-Hilbert problem. We compute asymptotic behaviours of these functions as $rx \to 0_{+}$ and $(r-1)x \to \infty$, as well large $n$ asymptotics for the associated Hankel determinants in several regimes of $r$ and $x$.