Universality in Complex Wishart ensembles: The 1 cut case

TL;DR

This paper investigates the universality of eigenvalue distributions in Wishart ensembles with two distinct eigenvalues, focusing on the case where the spectrum support is a single interval, revealing sine, Airy, and Tracy-Widom behaviors.

Contribution

It extends previous work by analyzing the single-interval support case using Riemann-Hilbert methods, demonstrating universal eigenvalue statistics.

Findings

Eigenvalue correlation kernel converges to sine and Airy kernels

Largest eigenvalue follows Tracy-Widom distribution

Supports universality in Wishart ensembles with two eigenvalues

Abstract

We studied universality of Wishart ensembles whose covariance matrix has 2 distinct eigenvalues and the number of each of these eigenvalue goes to infinity in the asymptotic limit. In this case, the limiting eigenvalue distribution can be supported on 1 or 2 disjoint intervals. In our previous work the case when the support consists of 2 intervals was studied. This paper complements our previous analysis and studied the case when the support consists of a single interval. 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.