Compact support probability distributions in random matrix theory

TL;DR

This paper generalizes fixed and bounded trace ensembles in random matrix theory with polynomial potentials, proving their equivalence in the large-N limit and deriving explicit eigenvalue distributions, including finite support cases.

Contribution

It introduces a generalized framework for fixed and bounded trace ensembles with polynomial potentials and establishes their equivalence in the large-N limit, providing explicit eigenvalue distribution results.

Findings

Fixed and bounded trace ensembles are equivalent in the large-N limit.

Eigenvalue distribution matches that of the canonical ensemble with measure exp[-n Tr V(M)].

Derived exact finite-N expressions for one- and two-point correlators with finite support.

Abstract

We consider a generalization of the fixed and bounded trace ensembles introduced by Bronk and Rosenzweig up to an arbitrary polynomial potential. In the large-N limit we prove that the two are equivalent and that their eigenvalue distribution coincides with that of the "canonical" ensemble with measure exp[-$n$Tr V(M)]. The mapping of the corresponding phase boundaries is illuminated in an explicit example. In the case of a Gaussian potential we are able to derive exact expressions for the one- and two-point correlator for finite $n$, having finite support.