Universal Cubic Eigenvalue Repulsion for Random Normal Matrices

TL;DR

This paper demonstrates that cubic eigenvalue repulsion in the complex plane is a universal feature of normal matrix models, providing analytical and numerical insights into eigenvalue distributions and correlations.

Contribution

It introduces the universality of cubic eigenvalue repulsion in normal matrices and derives analytical expressions for eigenvalue density and correlations.

Findings

Eigenvalue density and correlation functions are analytically derived.

Numerical eigenvalue statistics agree with analytical predictions.

Normal matrix models are more analytically tractable than complex matrix models.

Abstract

Random matrix models consisting of normal matrices, defined by the sole constraint $[N^{\dag},N]=0$, will be explored. It is shown that cubic eigenvalue repulsion in the complex plane is universal with respect to the probability distribution of matrices. The density of eigenvalues, all correlation functions, and level spacing statistics are calculated. Normal matrix models offer more probability distributions amenable to analytical analysis than complex matrix models where only a model wth a Gaussian distribution are solvable. The statistics of numerically generated eigenvalues from gaussian distributed normal matrices are compared to the analytical results obtained and agreement is seen.