  In this note we prove convergence of Green functions with Neumann boundary
conditions for the random walk to their continuous counterparts. Also a few
Beurling type hitting estimates are obtained for the random walk on
discretizations of smooth domains. These have been used recently in the study
of a two dimensional competing aggregation system known as $Competitive\,
Erosion$. Some of the statements appearing in this note are classical for
${\mathbb{Z}}^2$. However additional arguments are needed for the proofs in the
bounded geometry setting.
