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

- Authors
- Publication Date
- Jan 01, 1998
- Identifiers
- DOI: 10.1103/PhysRevLett.81.248
- OAI: oai:cds.cern.ch:342376
- Source
- CERN Document Server
- Keywords
- Language
- English
- License
- Unknown
- External links

## Abstract

In this paper the kernel for 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 Brézin and Neuberger. Microscopic universality for all three chiral ensembles then follows from universal behavior for $\beta =2$ both at the hard edge (shown by Akemann, Damgaard, Magnea and Nishigaki) and at the soft edge of the spectrum (shown by Kanzieper and Freilikher).