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

Details Universality in Chiral Random Matrix Theory at $β=1$ and $β=4$ Universality in Chiral Random Matrix Theory at $β=1$ and $β=4$

1998-01-08

arxiv.org/abs/hep-th/9801042v2

Phys.Rev.Lett. 81 (1998) 248-251

Physics

High Energy Physics

High Energy Physics - Theory

4 pages, modified discussion of edge contributions and corrected typos

Scientific paper

10.1103/PhysRevLett.81.248

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.

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