Large Complex Correlated Wishart Matrices: The Pearcey Kernel and Expansion at the Hard Edge

TL;DR

This paper analyzes the eigenvalue behavior of large complex correlated Wishart matrices near cusp points and the hard edge, revealing new universal kernels and correction formulas under mild assumptions.

Contribution

It provides a detailed description of eigenvalue fluctuations near cusp points using the Pearcey kernel and refines the understanding of the hard edge behavior with explicit correction terms.

Findings

Eigenvalue density vanishes like a cube root at cusp points.

Eigenvalue fluctuations near cusp points are described by the Pearcey kernel.

Explicit $1/N$ correction term for the smallest eigenvalue fluctuation.

Abstract

We study the eigenvalue behaviour of large complex correlated Wishart matrices near an interior point of the limiting spectrum where the density vanishes (cusp point), and refine the existing results at the hard edge as well. More precisely, under mild assumptions for the population covariance matrix, we show that the limiting density vanishes at generic cusp points like a cube root, and that the local eigenvalue behaviour is described by means of the Pearcey kernel if an extra decay assumption is satisfied. As for the hard edge, we show that the density blows up like an inverse square root at the origin. Moreover, we provide an explicit formula for the $1/N$ correction term for the fluctuation of the smallest random eigenvalue.