  We obtain asymptotic expansions of the spatially discrete 2D heat kernels, or
Green's functions on lattices, with respect to powers of time variable up to an
arbitrary order and estimate the remainders uniformly on the whole lattice.
Unlike in the 1D case, the asymptotics contains a time independent term. The
derivation of its spatial asymptotics is the technical core of the paper.
Besides numerical applications, the obtained results play a crucial role in the
analysis of spatio-temporal patterns for reaction-diffusion equations on
lattices, in particular rattling patterns for hysteretic diffusion systems.
