Affordable Access

PDEs satisfied by extreme eigenvalues distributions of GUE and LUE

Authors
Type
Preprint
Publication Date
Submission Date
Source
arXiv
License
Yellow
External links

Abstract

In this paper we study, $\textsf{Prob}(n,a,b),$ 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 $\textsf{Prob}(n,a,b)$, namely, $$ H_n(a,b):=\left[\frac{\partial}{\partial a}+\frac{\partial}{\partial b}\right]\ln\textsf{Prob}(n,a,b),$$ in the Gaussian Unitary Ensemble (GUE) and $$ H_n(a,b):=\left[a\frac{\partial}{\partial a}+b\frac{\partial}{\partial b}\right]\ln \textsf{Prob}(n,a,b),$$ 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 Painlev\'{e} IV and a Painlev\'{e} 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.

Statistics

Seen <100 times