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

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 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.

Critical Edge Behavior and the Bessel to Airy Transition in the Singularly Perturbed Laguerre Unitary Ensemble

“…[10,19,30], that the eigenvalue correlation kernel for the ensemble (1.1) has the following form K n (x, y; t) = x α 2 y α 2 e − V t (x)+V t (y) 2 n−1 k=0 p k (x)p k (y), (1.4) where p k (x) denotes the k-th degree orthonormal polynomial with respect to the weight w(x) = w(x; t) = x α e −Vt(x) , x ∈ (0, ∞), t > 0, α > 0. (1.5) Using the famous Christoffel-Darboux formula, (1.4) can be put into the following closed form K n (x, y; t) = γ 2 n−1 w(x)w(y) π n (x)π n−1 (y) − π n−1 (x)π n (y) 6) where γ k is the leading coefficient of p k (x), and π k (x) is the monic polynomial such that p k (x) = γ k π k (x). In the study of random matrices, there is a lot of interest in the limit of the correlation kernel K n (x, y) when the matrix size n tends to infinity.…”

“…In fact, Chen and Its [6] use (1.5) as a concrete and important example of the Pollaczek-Szegö type orthogonal polynomials, supported on infinite intervals. The Hankel determinant, which is the normalizing constant Z n in (1.2), plays a fundamental role in [6], upon which the main results are derived and stated. A relation is also found between Hankel determinant and the Jimbo-Miwa-Ueno isomonodromy τ -function.…”

“…As is shown in Chen and Its [6], various quantities, such as the three-term recurrence coefficients of the associated orthogonal polynomials, are expressed in terms of a specific solution to a PIII equation. On the other hand, in the present paper, the previous derivation indicates that the third-order equation would play the same role.…”

Perturbed Hankel determinant, correlation functions and Painlevé equations

“…First, the monic orthogonal polynomials P n (x; t, α) with respect to the singular perturbed Laguerre weight, w(x; t, α) = x α e −x e − t x , 0 ≤ x < ∞, α > 0, t > 0, satisfy the ladder operator relations have been derived in [18] , which we restate here,…”

“…The monic orthogonal polynomials P n (x; t, α) satisfy ladder operator relations in t , see (5.56) and (5.57) in [18], d dt − r n xt P n (x; t, α) = − β n R n xt P n−1 (x; t, α),…”