New Multicritical Random Matrix Ensembles

TL;DR

This paper introduces a new family of random matrix ensembles characterized by a parameter alpha, revealing a continuum of universality classes with unique eigenvalue density and spacing behaviors near zero.

Contribution

It constructs a novel class of random matrix ensembles with a continuous parameter, expanding the understanding of universality classes in eigenvalue statistics.

Findings

Eigenvalue density near zero scales as |x|^α.

Eigenvalue spacing near zero scales as 1/N^{1/(1+α)}.

Derived formulas for eigenvalue density and approximate scaling functions.

Abstract

In this paper we construct a class of random matrix ensembles labelled by a real parameter $\alpha \in (0,1)$, whose eigenvalue density near zero behaves like $|x|^\alpha$. The eigenvalue spacing near zero scales like $1/N^{1/(1+\alpha)}$ and thus these ensembles are representatives of a {\em continous} series of new universality classes. We study these ensembles both in the bulk and on the scale of eigenvalue spacing. In the former case we obtain formulas for the eigenvalue density, while in the latter case we obtain approximate expressions for the scaling functions in the microscopic limit using a very simple approximate method based on the location of zeroes of orthogonal polynomials.