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

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 View the MathML sourcew(x)=w(x,s)≔xαe−xe−s/x,0≤x<∞,α>0,s>0, Turn MathJax on namely, the determination of the associated Hankel determinant and recurrence coefficients. Here w(x,s)w(x,s) is the Laguerre weight View the MathML sourcexαe−x perturbed by a multiplicative factor View the MathML sourcee−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 View the MathML source{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.