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

In this paper the kernel for the spectral correlation functions of the invariant chiral random matrix ensembles with real ($β=1$) and quaternion real ($β= 4$) matrix elements is expressed in terms of the kernel of the corresponding complex Hermitean random matrix ensembles ($β=2$). 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. Universal behavior at the hard edge of the spectrum for all three chiral ensembles then follows from microscopic universality for $β=2$ as shown by Akemann, Damgaard, Magnea and Nishigaki.

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