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

Yang Chen; Alexander Its

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arXiv · 0808.3590

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

Yang Chen·Alexander Its

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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 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 $\{x_j:j=1,..,,n\};$ $\sum_{j=1}^{n}1/x_j.$ 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 Painlevé function.

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Yang Chen, Alexander Its. 2009-02-04. Painléve III and a singular linear statistics in Hermitian random matrix ensembles I. https://arxiv.org/abs/0808.3590

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