Thouless Energy and Correlations of QCD Dirac Eigenvalues

Eigenvalues and eigenfunctions of the QCD Dirac operator are studied for an instanton liquid partition function. We find that for energy differences $\delta E$ below an energy scale $E_c$, identified as the Thouless energy, the eigenvalue correlations are given by random matrix theory. The value of $E_c$ shows a weak volume dependence for eigenvalues near zero and is consistent with a scaling of $E_c\sim 1/L^2$ in the bulk of the spectrum in agreement with estimates from chiral perturbation theory that $E_c/\Delta \approx F_{\pi }^2{L}^2/\pi$ (with average level spacing $\Delta$). For $\delta E>\mathrm{}{E}_c$ the number variance shows a linear dependence. For the wave functions we find a small nonzero multifractality index.