  Lieb-Robinson-type bounds are reported for a large class of classical
Hamiltonian lattice models. By a suitable rescaling of energy or time, such
bounds can be constructed for interactions of arbitrarily long range. The bound
quantifies the dependence of the system's dynamics on a perturbation of the
initial state. The effect of the perturbation is found to be effectively
restricted to the interior of a causal region of logarithmic shape, with only
small, algebraically decaying effects in the exterior. A refined bound, sharper
than conventional Lieb-Robinson bounds, is required to correctly capture the
shape of the causal region, as confirmed by numerical results for classical
long-range $XY$ chains. We discuss the relevance of our findings for the
relaxation to equilibrium of long-range interacting lattice models.
