Spectral Properties of the Wilson Dirac Operator and random matrix theory

Random Matrix Theory has been successfully applied to lattice Quantum Chromodynamics. In particular, a great deal of progress has been made on the understanding, numerically as well as analytically, of the spectral properties of the Wilson Dirac operator. In this paper, we study the infra-red spectrum of the Wilson Dirac operator via Random Matrix Theory including the three leading order $a^2$ correction terms that appear in the corresponding chiral Lagrangian. A derivation of the joint probability density of the eigenvalues is presented. This result is used to calculate the density of the complex eigenvalues, the density of the real eigenvalues and the distribution of the chiralities over the real eigenvalues. A detailed discussion of these quantities shows how each low energy constant affects the spectrum. Especially we consider the limit of small and large (which is almost the mean field limit) lattice spacing. Comparisons with Monte Carlo simulations of the Random Matrix Theory show a perfect agreement with the analytical predictions. Furthermore we present some quantities which can be easily used for comparison of lattice data and the analytical results.