  The paper is devoted to understand the large time behaviour and decay of the
solution of the discrete heat equation in the one dimensional mesh $\Z$ on
$\ell^p$ spaces, and its analogies with the continuous-space case. We do a deep
study of the moments of the discrete gaussian kernel (which is given in terms
of Bessel functions), in particular the mass conservation principle; that is
reflected on the large time behaviour of solutions. We prove asymptotic
pointwise and $\ell^p$ decay results for the fundamental solution. We use that
estimates to get rates on the $\ell^p$ decay and large time behaviour of
solutions. For the $\ell^2$ case, we get optimal decay by use of Fourier
techniques.
