Family of solvable generalized random-matrix ensembles with unitary symmetry

TL;DR

This paper introduces a broad family of unitary-invariant random matrix ensembles characterized by a spread function, enabling the modeling of diverse eigenvalue distributions and critical phenomena beyond classical Gaussian ensembles.

Contribution

It constructs a general class of solvable random matrix ensembles with a novel spread function, extending the scope of models with unitary symmetry.

Findings

Exact correlation functions for generalized ensembles are derived.

Different spread functions can produce arbitrary eigenvalue densities.

The framework includes critical ensembles with multifractality.

Abstract

We construct a very general family of characteristic functions describing Random Matrix Ensembles (RME) having a global unitary invariance, and containing an arbitrary, one-variable probability measure which we characterize by a `spread function'. Various choices of the spread function lead to a variety of possible generalized RMEs, which show deviations from the well-known Gaussian RME originally proposed by Wigner. We obtain the correlation functions of such generalized ensembles exactly, and show examples of how particular choices of the spread function can describe ensembles with arbitrary eigenvalue densities as well as critical ensembles with multifractality.