Universality in Chiral Random Matrix Theory at $\beta =1$ and $\beta =4$

TL;DR

This paper establishes exact and asymptotic identities connecting spectral correlation kernels of chiral random matrix ensembles with real and quaternion elements to those with complex Hermitean matrices, demonstrating universality at the spectrum's hard edge.

Contribution

It provides new exact and asymptotic relations between different beta ensembles' kernels, extending universality results to $eta=1$ and $eta=4$ cases.

Findings

Exact kernel identities for Gaussian distributions

Asymptotic kernel identities for polynomial potentials

Universal behavior at the spectrum's hard edge

Abstract

In this paper the kernel for the spectral correlation functions of the invariant chiral random matrix ensembles with real ($\beta =1$) and quaternion real ($\beta = 4$) matrix elements is expressed in terms of the kernel of the corresponding complex Hermitean random matrix ensembles ($\beta=2$). Such identities are exact in case of a Gaussian probability distribution and, under certain smoothness assumptions, they are shown to be valid asymptotically for an arbitrary finite polynomial potential. They are proved by means of a construction proposed by Br\'ezin and Neuberger. Universal behavior at the hard edge of the spectrum for all three chiral ensembles then follows from microscopic universality for $\beta =2$ as shown by Akemann, Damgaard, Magnea and Nishigaki.