Application of the τ-Function Theory of Painlevé Equations to Random Matrices: Pv, PIII, the LUE, JUE, and CUE

Abstract

With 〈·〉 denoting an average with respect to the eigenvalue PDF for the Laguerre unitary ensemble, the object of our study is ẼN (I;a,μ):=〈Πl=1N χ 0,∞)I(l)(λ-λl)μ〉 for I = (0, s) and I = (s, ∞), where χ = 1 for λl ∼ I and χ(l) = 0 otherwise. Using Okamoto's development of the theory of the Painlevé V equation, it is shown that Ẽ N (I; a, μ) is a τ-function associated with the Hamiltonian therein, and so can be characterized as the solution of a certain second-order second-degree differential equation, or in terms of the solution of certain difference equations. The cases μ. = 0 and μ = 2 are of particular interest, because they correspond to the cumulative distribution and density function, ..