Universal Fluctuations in Spectra of the Lattice Dirac Operator

Recently, Kalkreuter obtained the complete Dirac spectrum for an SU(2) lattice gauge theory. We performed a statistical analysis of his data and found that the eigenvalue correlations can be described by the Gaussian symplectic ensemble. Long range fluctuations are strongly suppressed: The variance of a sequence of levels containing $n$ eigenvalues on average is given by $(1/2{\ensuremath{\pi}}^{2})(\mathrm{ln}n+\mathrm{const})$. Our findings are in agreement with the antiunitary symmetry of the lattice Dirac operator for ${N}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$ with staggered fermions. For ${N}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3$ we predict that the eigenvalue correlations are given by the Gaussian unitary ensemble.

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