Applications in random matrix theory of a PIII$'$ $\tau$-function sequence from Okamoto's Hamiltonian formulation

TL;DR

This paper links a specific Toeplitz determinant related to Laguerre ensemble statistics to Painlevé III' equations, revealing new integrable structures and connections to random matrix theory and Wigner time delay distribution.

Contribution

It demonstrates that a Toeplitz determinant involving linear combinations of Bessel functions forms a $ au$-function sequence in Okamoto's Painlevé III' framework, providing new insights into its differential equations.

Findings

The logarithmic derivative satisfies the Painlevé III' $ au$-form.

The determinant relates to the distribution of the Wigner time delay.

Connections between generating functions and large-$n$ limits are clarified.

Abstract

We consider the singular linear statistic of the Laguerre unitary ensemble consisting of the sum of the reciprocal of the eigenvalues. It is observed that the exponential generating function for this statistic can be written as a Toeplitz determinant with entries given in terms of particular $K$ Bessel functions. Earlier studies have identified the same determinant, but with the $K$ Bessel functions replaced by $I$ Bessel functions, as relating to the hard edge scaling limit of a generalized gap probability for the Laguerre unitary ensemble, in the case of non-negative integer Laguerre parameter. We show that the Toeplitz determinant formed from an arbitrary linear combination of these two Bessel functions occurs as a $\tau$-function sequence in Okamoto's Hamiltonian formulation of Painlev\'e III$'$, and consequently the logarithmic derivative of both Toeplitz determinants satisfies the same $\sigma$-form Painlev\'e III$'$ differential equation, giving an explanation of a fact which can be observed from earlier results. In addition, some insights into the relationship between this characterization of the generating function, and its characterization in the $n \to \infty$ limit, both with the Laguerre parameter $\alpha$ fixed, and with $\alpha = n$ (this latter circumstance being relevant to an application to the distribution of the Wigner time delay statistic), are given.