Boundary conditions associated with the Painlevé III′ and v evaluations of some random matrix averages

Abstract

In a previous work a random matrix average for the Laguerre unitary ensemble, generalizing the generating function for the probability that an interval (0, s) at the hard edge contains k eigenvalues, was evaluated in terms of a Painlevé V transcendent in σ-form. However, the boundary conditions for the corresponding differential equation were not specified for the full parameter space. Here this task is accomplished in general, and the obtained functional form is compared against the most general small s behaviour of the Painlevé V equation in σ-form known from the work of Jimbo. An analogous study is carried out for the hard edge scaling limit of the random matrix average, which we have pre..