A Survey on the Eigenvalues Local Behavior of Large Complex Correlated Wishart Matrices

TL;DR

This survey reviews recent research on the local eigenvalue behavior of large complex correlated Wishart matrices, highlighting universal fluctuation patterns at spectrum edges and cusp points, and their implications for matrix condition numbers.

Contribution

It synthesizes recent findings on eigenvalue fluctuations at spectrum edges and cusp points, including new asymptotic results and independence properties.

Findings

Eigenvalue fluctuations at spectrum edges follow universal kernels (Airy, Bessel, Pearcey).

Eigenvalues at different soft edges are asymptotically independent.

Next order asymptotics at the hard edge are derived.

Abstract

The aim of this note is to provide a pedagogical survey of the recent works by the authors ( arXiv:1409.7548 and arXiv:1507.06013) concerning the local behavior of the eigenvalues of large complex correlated Wishart matrices at the edges and cusp points of the spectrum: Under quite general conditions, the eigenvalues fluctuations at a soft edge of the limiting spectrum, at the hard edge when it is present, or at a cusp point, are respectively described by mean of the Airy kernel, the Bessel kernel, or the Pearcey kernel. Moreover, the eigenvalues fluctuations at several soft edges are asymptotically independent. In particular, the asymptotic fluctuations of the matrix condition number can be described. Finally, the next order term of the hard edge asymptotics is provided.