Application of the τ-function theory of Painleve equations to random matrices: PIV, PII and the GUE

Tracy and Widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite Gaussian unitary ensemble (GUE) in terms of a Painleve IV and Painleve II transcendent respectively. We generalise these results to the evaluation of EN(λ;a) :(Πnl=1 χ(l)(-∞,λ)(λ - λl)a), where χ(l)(-∞) = 1 for λl E (- ∞,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) :(ΠNl=1(λ - λl)a). Of particular interest are EN(λ;2) and FN(λ;2)X 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 Painleve IV and Painleve II equations, 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 EN(λ; a) and FN(λ; a) for the other classical matrix ensembles