Level Crossing in Random Matrices: I. Random perturbation of a fixed matrix

TL;DR

This paper investigates the probability distribution of level crossing points in complex plane for a matrix family with a fixed matrix perturbed by Gaussian random matrices, providing exact and asymptotic formulas.

Contribution

It introduces new formulas for the distribution of level crossing points in matrix families with Gaussian perturbations, advancing understanding of spectral behavior under randomness.

Findings

Derived exact formulas for level crossing probabilities.

Provided asymptotic approximations for large matrices.

Analyzed the distribution of crossing points in the complex plane.

Abstract

We consider level crossing in a matrix family $H=H_0+\lambda V$ where $H_0$ is a fixed $N\times N$ matrix and $V$ belongs to one of the standard Gaussian random matrix ensembles. We study the probability distribution of level crossing points in the complex plane of $\lambda$, for which we obtain a number of exact, asymptotic and approximate formulas.