Painlevé III and a singular linear statistics in Hermitian random matrix ensembles, I

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)≔xαe−xe−s/x,0≤x<∞,α>0,s>0, namely, the determination of the associated Hankel determinant and recurrence coefficients. Here w(x,s)w(x,s) is the Laguerre weight xαe−x perturbed by a multiplicative factor e−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 Pollaczek and Szegö many years ago. Such weights are said to be singular or irregular due to the violation of the Szegö condition.In our problem, the linear statistics is a sum of the reciprocal of positive random variables {xj:j=1,…,n};∑j=1n1/xj.We show that the moment generating function, or the Laplace transform of the probability density function of this linear statistics, can be expressed as the ratio of Hankel determinants and as an integral involving a particular third Painlevé function.