Universality in chiral random matrix theory at β=1 and β=4

In this paper the kernel for the spectral correlation functions of 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 Hermitian 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 of the eigenvalues close to zero for all three chiral ensembles then follows from microscopic universality for β=2 as shown by Akemann, Damgaard, Magnea, and Nishigaki.