  In this paper we investigate the asymptotic behavior and decay of the
solution of the discrete in time $N$-dimensional heat equation. We give a
convergence rate with which the solution tends to the discrete fundamental
solution, and the asymptotic decay, both in $L^p(\R^N).$ Furthermore we prove
optimal $L^2$-decay of solutions. Since the technique of energy methods is not
applicable, we follow the approach of estimates based on the discrete
fundamental solution which is given by an original integral representation and
also by MacDonald's special functions. As a consequence, the analysis is
different to the continuous in time heat equation and the calculations are
rather involved.
