Progress on the Microscopic Spectrum of the Dirac Operator for QCD with\n Wilson Fermions

Starting from the chiral Lagrangian for Wilson fermions at nonzero lattice\nspacing we have obtained compact expressions for all spectral correlation\nfunctions of the Hermitian Wilson Dirac operator in the $\\epsilon$-domain of\nQCD with dynamical quarks. We have also obtained the distribution of the\nchiralities over the real eigenvalues of the Wilson Dirac operator for any\nnumber of flavors. All results have been derived for a fixed index of the Dirac\noperator. An important effect of dynamical quarks is that they completely\nsuppress the inverse square root singularity in the spectral density of the\nHermitian Wilson Dirac operator. The analytical results are given in terms of\nan integral over a diffusion kernel for which the square of the lattice spacing\nplays the role of time. This approach greatly simplifies the expressions which\nwe here reduce to the evaluation of two-dimensional integrals.\n