Painl\'eve III and a singular linear statistics in Hermitian random matrix ensembles I

TL;DR

This paper investigates a singular linear statistic in unitary Laguerre ensembles, linking it to orthogonal polynomials with a perturbed weight and expressing its moment generating function via Hankel determinants and Painlevé functions.

Contribution

It characterizes the Hankel determinant and recurrence coefficients for orthogonal polynomials with a singular weight, connecting the linear statistic to Painlevé equations.

Findings

Expressed the moment generating function as a ratio of Hankel determinants.

Connected the linear statistic to a third Painlevé function.

Analyzed the impact of the strong zero at the origin on the orthogonal polynomials.

Abstract

In this paper, we study a certain linear statistics of the unitary Laguerre ensembles, motivated in part by an integrable quantum field theory at finite temperature. It transpires that this is equivalent to the characterization of a sequence of polynomials orthogonal with respect to the weight $$ w(x)=w(x,s):=x^{\al}\rme^{-x}\rme^{-s/x}, \quad 0\leq x<\infty, \al>0, s>0, $$ namely, the determination of the associated Hankel determinant and recurrence coefficients. Here $w(x,s)$ is the Laguerre weight $x^{\al}\:\rme^{-x}$ 'perturbed' by a multiplicative factor $\rme^{-s/x},$ which induces an infinitely strong zero at the origin. For polynomials orthogonal on the unit circle, a particular example where there are explicit formulas, the weight of which has infinitely strong zeros, was investigated by Pollazcek and Szeg\"o many years ago. Such weights are said to be 'singular' or irregular due to the violation of the Szeg\"o condition. In our problem, the linear statistics is a sum of the reciprocal of positive random variables $\{x_j:j=1,..,,n\};$ $\sum_{j=1}^{n}1/x_j.$ We show that the moment generating function, or the Laplace transform of the probability density function of this linear statistics is expressed as the ratio of Hankel determinants and as an integral of the combination of a particular third Painlev\'e function.