Asymptotic correlations for Gaussian and Wishart matrices with external source

TL;DR

This paper analyzes how finite rank perturbations affect the correlation kernels of Gaussian and Wishart matrix ensembles, deriving asymptotic formulas and limiting kernels at spectral edges.

Contribution

It provides explicit formulas for the difference in kernels due to finite rank perturbations and computes their asymptotics at spectral edges for large matrices.

Findings

Difference between perturbed and unperturbed kernels is a degenerate kernel of size r.

Asymptotic formulas for Laguerre kernels involve Bessel and Airy functions.

Large N limits yield kernels at the spectrum's hard and soft edges.

Abstract

We consider ensembles of Gaussian (Hermite) and Wishart (Laguerre) $N\times N$ hermitian matrices. We study the effect of finite rank perturbations of these ensembles by a source term. The rank $r$ of the perturbation corresponds to the number of non-null eigenvalues of the source matrix. In the perturbed ensembles, the correlation functions can be written in terms of kernels. We show that for all $N$, the difference between the perturbed and the unperturbed kernels is a degenerate kernel of size $r$ which depends on multiple Hermite or Laguerre functions. We also compute asymptotic formulas for the multiple Laguerre functions kernels in terms multiple Bessel (resp. Airy) functions. This leads to the large $N$ limiting kernels at the hard (resp. soft) edge of the spectrum of the perturbed Laguerre ensemble. Similar results are obtained in the Hermite case.