New correlation functions for random matrices and integrals over supergroups

TL;DR

This paper develops new correlation functions for random matrices using supergroup integrals, deriving key spectral statistics for classical ensembles and exploring universality with external sources.

Contribution

It introduces an extension of Harish-Chandra-Itzykson-Zuber integrals to supergroups, enabling explicit evaluation of matrix averages in random matrix theory.

Findings

Derived density of states and two-point correlation functions.

Extended integrals to GOE ensemble and supergroups.

Discussed universality with external matrix sources.

Abstract

The averages of ratios of characteristic polynomials det(lambda - X) of N x N random matrices X, are investigated in the large N limit for the GUE, GOE and GSE ensemble. The density of states and the two-point correlation function are derived from these ratios. The method relies on an extension of the Harish-Chandra-Itzykson-Zuber integrals to the GOE ensemble and to supergroups, which are explicitly evaluated as solutions of heat kernel differential equations. An external matrix source, linearly coupled to the random matrices, may also be added to the Gaussian distribution, and allows for a discussion of universality of the GOE results in the large N limit.