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 weightnamely, the determination of the associated Hankel determinant and recurrence coefficients. Here w(x, s) is the Laguerre weight x α e −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 {x j : j = 1, . . . , n}; n j=1 1/x j . 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.
We consider an exactly solvable random matrix model related to the random transfer matrix model for disordered conductors. In the conventional random matrix models the spacing distribution of nearest neighbor eigenvalues, when expressed in units of average spacing, has a universal behavior known generally as the Wigner distribution. In contrast, our model has a single parameter, as a function of which the spacing distribution crosses over from a Wigner to a distribution which is increasingly more Poissonlike, a feature common to a wide variety of physical systems including disorder and chaos.
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