Random matrix models with log-singular level confinement: method of fictitious fermions

TL;DR

This paper develops a formalism using fictitious fermions to analyze eigenvalue correlations in non-Gaussian random matrix ensembles with log-singular confinement, deriving universal correlation expressions.

Contribution

It introduces a new method based on fictitious fermions and an effective Schrödinger equation to study complex eigenvalue correlations in non-monotonic, log-singular ensembles.

Findings

Derived analytical two-point kernel expressions in various spectral limits.

Predicted universal correlations near the spectrum endpoint.

Established a connection between eigenvalue correlations and Dyson's density of states.

Abstract

Joint distribution function of N eigenvalues of U(N) invariant random-matrix ensemble can be interpreted as a probability density to find N fictitious non-interacting fermions to be confined in a one-dimensional space. Within this picture a general formalism is developed to study the eigenvalue correlations in non-Gaussian ensembles of large random matrices possessing non-monotonic, log-singular level confinement. An effective one-particle Schroedinger equation for wave-functions of fictitious fermions is derived. It is shown that eigenvalue correlations are completely determined by the Dyson's density of states and by the parameter of the logarithmic singularity. Closed analytical expressions for the two-point kernel in the origin, bulk, and soft-edge scaling limits are deduced in a unified way, and novel universal correlations are predicted near the end point of the single spectrum support.