  We investigate the eigenvalue spectrum of the staggered Dirac matrix in SU(3)
gauge theory and in full QCD as well as in quenched U(1) theory on various
lattice sizes. As a measure of the fluctuation properties of the eigenvalues,
we consider the nearest-neighbor spacing distribution, $P(s)$. We further study
two-color QCD at nonzero chemical potential, $\mu$, by constructing the spacing
distribution of adjacent eigenvalues in the complex plane. We find that in all
regions of their phase diagrams, compact lattice gauge theories have bulk
spectral correlations given by random matrix theory, which is an indication for
quantum chaos. In the confinement phase, the low-lying Dirac spectrum of these
quantum field theories is well described by random matrix theory, exhibiting
universal behavior.
