Some Universal Properties for Restricted Trace Gaussian Orthogonal, Unitary and Symplectic Ensembles

TL;DR

This paper establishes universal limits of correlation functions at zero, the spectrum edge, and in the bulk for fixed and bounded trace Gaussian orthogonal, unitary, and symplectic ensembles, extending known universality results.

Contribution

It proves universality of correlation functions for restricted trace Gaussian ensembles at multiple spectral regions, including new results for bounded trace Gaussian unitary ensembles.

Findings

Universal limits at zero spectrum

Universal limits at spectrum edge

Universal limits in the bulk for bounded trace Gaussian unitary ensemble

Abstract

Consider fixed and bounded trace Gaussian orthogonal, unitary and symplectic ensembles, closely related to Gaussian ensembles without any constraint. For three restricted trace Gaussian ensembles, we prove universal limits of correlation functions at zero and at the edge of the spectrum edge. In addition, by using the universal result in the bulk for fixed trace Gaussian unitary ensemble, which has been obtained by G$\ddot{o}$tze and Gordin, we also prove universal limits of correlation functions for bounded trace Gaussian unitary ensemble.