  For parabolic spatially discrete equations, we consider Green's functions,
also known as heat kernels on lattices. We obtain their asymptotic expansions
with respect to powers of time variable $t$ up to an arbitrary order and
estimate the remainders uniformly on the whole lattice. The spatially discrete
(difference) operators under consideration are finite-difference approximations
of continuous strongly elliptic differential operators (with constant
coefficients) of arbitrary even order in $\mathbb R^d$ with arbitrary
$d\in\mathbb N$. This genericity, besides numerical and deterministic
lattice-dynamics applications, allows one to obtain higher-order asymptotics of
transition probability functions for continuous-time random walks on $\mathbb
Z^d$ and other lattices.
