The Hankel determinant associated with a singularly perturbed Laguerre unitary ensemble

Abstract

We are concerned with the probability that all the eigenvalues of a unitary ensemble with the weight function \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$w(x;1) = {x^\\alpha }{e^{ - x - {1 \\over x}}},x \\in [0,\\infty )$$\\end{document}, α > −1, t ≥ 0, are greater than s. This probability is expressed as the quotient of Dn(s,t) and its value at s = 0, where Dn(s,t) denotes the determinant of the n dimensional Hankel matrices generated by the moments of w(x;t) on x ∈ [s,∞). In this paper we focus specifically on the Hankel determinant Dn(s,t) and its properties.Based on the ladder operators adapted to the monic polynomials orthogonal with respect to w(x;t), and from the associated supplementary conditions and a sum-rule, we show that the log-derivative of the Hankel determinant, viewed as a function of s and t, satisfies a second order sixth degree partial differential equation, where n appears as a parameter. In order to go to the thermodynamic limit, of infinitely large matrices, we envisage a scenario where n → ∞, s → 0, and t → 0 such that S := 4ns and T := (2n + 1 + α)t are finite. After such a double scaling, the large finite n equation reduces to a second order second degree equation, in the variables S and T, from which we derive the asymptotic expansion of the scaled Hankel determinant in three cases of S and T: S → ∞ with T fixed, S → 0 with T > 0 fixed, and T → ∞ with S > 0 fixed. The constant term in the asymptotic expansion is shown to satisfy a difference equation and one of its solutions is the Tracy-Widom constant.