Application of the τ-function theory of Painlevé equations to Random Matrices: PIV, PII and the GUE

Abstract

Tracy and Widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite GUE in terms of a PIV and PII transcendent respectively. We generalise these results to the evaluation of ẼN(λ; a):= <ΠNl=1 χ(l) (-∞,λ](λ - λl)a>, where χ(l) (-∞,λ] = 1 for λl ∈ (-∞, λ] and χ(l) (-∞,λ] = 0 otherwise, and the average is with respect to the joint eigenvalue distribution of the GUE, as well as to the evaluation of FN(λ; a):= <ΠN l=1(λ - λl)a>. Of particular interest are ẼN(λ; 2) and FN (λ; 2), and their scaled limits, which give the distribution of the largest eigenvalue and the density respectively. Our results are obtained by applying the Okamoto τ-function ..