Edge effects in some perturbations of the GUE

TL;DR

This paper analyzes eigenvalue distributions of bordered GUE matrices with specific Gaussian perturbations, deriving explicit formulas and demonstrating how parameter tuning can induce eigenvalue separation, revealing universal correlation behavior.

Contribution

It provides explicit eigenvalue probability functions and correlation kernels for bordered GUE matrices with Gaussian perturbations, including large N limits and eigenvalue separation phenomena.

Findings

Explicit eigenvalue probability functions derived

Correlation kernels computed explicitly for single bordering case

Eigenvalue separation controlled by a universal parameter in large N limit

Abstract

A bordering of GUE matrices is considered, in which the bordered row consists of zero mean complex Gaussians N$[0,\sigma/2] + i {\rm N}[0,\sigma/2]$ off the diagonal, and the real Gaussian N$[\mu,\sigma/\sqrt{2}]$ on the diagonal. We compute the explicit form of the eigenvalue probability function for such matrices, as well as that for matrices obtained by repeating the bordering. The correlations are in general determinantal, and in the single bordering case the explicit form of the correlation kernel is computed. In the large $N$ limit it is shown that $\mu$ and/or $\sigma$ can be tuned to induce a separation of the largest eigenvalue. This effect is shown to be controlled by a single parameter, universal correlation kernel.