  Recently, Kalkreuter obtained the complete Dirac spectrum for an $SU(2)$
lattice gauge theory with dynamical staggered fermions on a $12^4$ lattice for
$\beta =1.8$ and $\beta=2.8$. We performed a statistical analysis of his data
and found that the eigenvalue correlations can be described by the Gaussian
Symplectic Ensemble. In particular, long range fluctuations are strongly
suppressed: the variance of a sequence of levels containing $n$ eigenvalues on
average is given by $\Sigma_2(n) \sim\frac 1{2\pi^2}(\log n + {\rm const.})$
instead of $\Sigma_2(n) = n$ for a random sequence of levels. Our findings are
in agreement with the anti-unitary symmetry of the lattice Dirac operator for
$N_c=2$ with staggered fermions which differs from the continuuum theory. For
$N_c = 3$ we predict that the eigenvalue correlations are given by the Gaussian
Unitary Ensemble.
