Level crossing in random matrices. II Random perturbation of a random matrix

TL;DR

This paper investigates the distribution of level crossings in spectra of linear matrix families with Gaussian ensembles, providing theoretical, numerical results, and insights into monodromy for 3x3 matrices.

Contribution

It introduces new theoretical and numerical analyses of level crossing distributions in Gaussian ensembles, including monodromy behavior for 3x3 matrices.

Findings

Distribution patterns for level crossings in Gaussian ensembles

Numerical results on monodromy in 3x3 matrix families

Theoretical formulations for spectral behavior

Abstract

In this paper we study the distribution of level crossings for the spectra of linear families A+lambda B, where A and B are square matrices independently chosen from some given Gaussian ensemble and lambda is a complex-valued parameter. We formulate a number of theoretical and numerical results for the classical Gaussian ensembles and some generalisations. Besides, we present intriguing numerical information about the distribution of monodromy in case of linear families for the classical Gaussian ensembles of 3 * 3 matrices.