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 , where for and otherwise, and the average is with respect to the joint eigenvalue distribution of the GUE, as well as to the evaluation of . Of particular interest are 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 theory of PIV and PII, for which we give a self contained presentation based on the recent work of Noumi and Yamada. We point out that the same approach can be used to study the quantities and FN(λ;a) for the other classical matrix ensembles.