Universality in Complex Wishart ensembles: The 2 cut case

TL;DR

This paper investigates the universality properties of Wishart matrices with two distinct eigenvalues, analyzing their spectral distribution and correlation kernels, and establishing conditions for disjoint spectral support.

Contribution

It provides a comprehensive analysis of the eigenvalue distribution and correlation kernels for Wishart ensembles with two eigenvalues, including conditions for disjoint spectral support and asymptotic behaviors.

Findings

Eigenvalue distribution supported on 1 or 2 intervals depending on parameters

Limiting correlation kernel given by sine and Airy kernels in bulk and edge

Largest eigenvalue follows Tracy-Widom distribution

Abstract

We studied the universality of Wishart ensembles whose covariance matrix has 2 distinct eigenvalues. We studied the asymptotic limit when the number of both eigenvalues goes to infinity and obtained universality results. In this case, the limiting eigenvalue distribution can be supported on 1 or 2 disjoint intervals. We obtained a necessary and sufficient condition on the parameters such that the limiting distribution is supported on 2 disjoint intervals and have computed the eigenvalue density in the limit. Furthermore, by using Riemann-Hilbert analysis, we have shown that under proper rescaling of the eigenvalues, the limiting correlation kernel is given by the sine kernel and the Airy kernel in the bulk and the edge of the spectrum respectively. As a consequence, the behavior of the largest eigenvalue in this model is described by the Tracy-Widom distribution.