Small eigenvalues of the SU(3) Dirac operator on the lattice and in random matrix theory

We calculate complete spectra of the staggered Dirac operator on the lattice in quenched SU(3) gauge theory for $\beta =5.4\mathrm{}$ and various lattice sizes. The microscopic spectral density, the distribution of the smallest eigenvalue, and the two-point spectral correlation function are analyzed. We find the expected agreement of the lattice data with universal predictions of the chiral unitary ensemble of random matrix theory up to a certain energy scale, the Thouless energy. The deviations from the universal predictions are determined using the disconnected scalar susceptibility. We find that the Thouless energy scales with the lattice size as expected from theoretical arguments making use of the Gell-Mann–Oakes–Renner relation.