Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edges

TL;DR

This paper analyzes the asymptotic fluctuations of eigenvalues at the edges of large complex correlated Wishart matrices, revealing Tracy-Widom fluctuations and asymptotic independence, with special results at the hard edge.

Contribution

It establishes the Tracy-Widom law for eigenvalue fluctuations at positive edges and proves asymptotic independence between extremal eigenvalues at different edges.

Findings

Eigenvalues at positive edges follow Tracy-Widom fluctuations.

Extremal eigenvalues at different edges are asymptotically independent.

Eigenvalues at the hard edge follow Bessel kernel fluctuations.

Abstract

We study the asymptotic behavior of eigenvalues of large complex correlated Wishart matrices at the edges of the limiting spectrum. In this setting, the support of the limiting eigenvalue distribution may have several connected components. Under mild conditions for the population matrices, we show that for every generic positive edge of that support, there exists an extremal eigenvalue which converges almost surely toward that edge and fluctuates according to the Tracy-Widom law at the scale $N^{2/3}$. Moreover, given several generic positive edges, we establish that the associated extremal eigenvalue fluctuations are asymptotically independent. Finally, when the leftmost edge is the origin (hard edge), the fluctuations of the smallest eigenvalue are described by mean of the Bessel kernel at the scale $N^2$.