  Recently, Kalkreuter obtained complete Dirac spectra for $SU(2)$ lattice
gauge theory both for staggered fermions and for Wilson fermions. The lattice
size was as large as $12^4$. We performed a statistical analysis of these data
and found that the eigenvalue correlations can be described by the Gaussian
Symplectic Ensemble for staggered fermions and by the Gaussian Orthogonal
Ensemble for Wilson fermions. In both cases long range spectral fluctuations
are strongly suppressed: the variance of a sequence of levels containing $n$
eigenvalues on average is given by $\Sigma_2(n) \sim 2 (\log n)/\beta\pi^2 $
($\beta$ is equal to 4 and 1, respectively) 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 Wilson fermions (with the continuum anti-unitary symmetry).
For $N_c = 3$, we predict that the eigenvalue correlations are given by the
Gaussian Unitary Ensemble.
