  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.
