Given a probability measure on a finitely generated group, its Martin
boundary is a way to compactify the group using the Green's function of the
corresponding random walk. We give a complete topological characterization of
the Martin boundary of finitely supported random walks on relatively hyperbolic
groups with virtually abelian parabolic subgroups. In particular, in the case
of nonuniform lattices in the real hyperbolic space H n , we show that the
Martin boundary coincides with the CAT (0) boundary of the truncated space, and
thus when n = 3, is homeomorphic to the Sierpinski carpet