PDEs SATISFIED BY EXTREME EIGENVALUES DISTRIBUTIONS OF GUE AND LUE

Abstract

In this paper we study, Prob (n, a, b), the probability that all the eigenvalues of finite n unitary ensembles lie in the interval (a, b). This is identical to the probability that the largest eigenvalue is less than b and the smallest eigenvalue is greater than a. It is shown that a quantity allied to Prob (n, a, b), namely, [Formula: see text] in the Gaussian Unitary Ensemble (GUE) and [Formula: see text], in the Laguerre Unitary Ensemble (LUE) satisfy certain nonlinear partial differential equations for fixed n, interpreting Hn(a, b) as a function of a and b. These partial differential equations may be considered as two variable generalizations of a Painlevé IV and a Painlevé V system, respectively. As an application of our result, we give an analytic proof that the extreme eigenvalues of the GUE and the LUE, when suitably centered and scaled, are asymptotically independent.