Random matrix triality at nonzero chemical potential

We introduce three universality classes of chiral random matrix ensembles with a nonzero chemical potential and real, complex or quaternion real matrix elements. In the thermodynamic limit we find that the distribution of the eigenvalues in the complex plane does not depend on the Dyson index, and is given by the solution proposed by Stephanov. For a finite number of degrees of freedom, $N,$ we find an accumulation of eigenvalues on the imaginary axis for real matrices, whereas for quaternion real matrices we find a depletion of eigenvalues in this domain. This effect is of order $1/\sqrt{N}.$ In particular for the real case the resolvent shows a discontinuity of order $1/\sqrt{N}.$ These results are in agreement with lattice QCD simulations with staggered fermions and recent instanton liquid simulations both for two colors and a nonzero chemical potential.