Convergence of discrete Green functions with Neumann boundary conditions

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.