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.

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Cited by in corpus (3)

• Limits on Sparse Data Acquisition: RIC Analysis of Finite Gaussian Matrices
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• A Survey on the Eigenvalues Local Behavior of Large Complex Correlated Wishart Matrices