Spectral Fluctuations of the QCD Dirac Operator

Recently, QCD Dirac spectra have been obtained for reasonably large lattices. We argue that correlations of these spectra are universal and can be obtained from a random matrix model with the global symmetries of QCD. Analytical arguments are presented to support this claim. Detailed comparisons with lattice results are made. We analyze spectral correlations in the bulk of the spectrum and find that long range fluctuations are strongly suppressed in complete agreement with random matrix theory. We study the valence quark mass dependence of the lattice QCD Dirac operator showing that the average spectral density near zero virtuality is given by chiral random matrix theory. This agrees with recent lattice calculations of the microscopic spectral density. of this lecture, we introduce a schematic random matrix model for the chiral phase transition at nonzero chemical potential. We consider both real, complex and quaternion real matrix elements. We find a remarkable triality for the distribution of of the eigenvalues in the complex plane. In particular, this elucidates the nature of the quenched approximation in each case.