PDEs satisfied by extreme eigenvalues distributions of GUE and LUE

Abstract

In this paper we study, $\text{Prob}(n,a,b),$ 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 $\text{Prob}(n,a,b)$, namely, $$H_n(a,b):=\left[\frac{\partial }{\partial a}+\frac{\partial }{\partial b}\right]\ln \text{Prob}(n,a,b),$$ in the Gaussian Unitary Ensemble (GUE) and $$H_n(a,b):=\left[a\frac{\partial }{\partial a}+b\frac{\partial }{\partial b}\right]\ln \text{Prob}(n,a,b),$$ in the Laguerre Unitary Ensemble (LUE) satisfy certain nonlinear partial differential equations for fixed $n$, interpreting $H_n(a,b)$ as a function of $a$ and $b$. These partial differential equations maybe 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.

References in corpus (3)

• Universality for mathematical and physical systems
• Perturbed Hankel Determinants: Applications to the Information Theory of MIMO Wireless Communications
• Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble