RANDOM WALKS ON CARTESIAN PRODUCTS OF CERTAIN NONAMENABLE GROUPS AND INTEGER LATTICES

A random walk on a discrete group satisfies a local limit theorem with power law exponent \alpha if the return probabilities follow the asymptotic law P{ return to starting point after n steps } ~ C \rho^n n^{-\alpha }. A group has a universal local limit theorem if all random walks on the group with finitely supported step distributions obey a local limit theorem with the same power law exponent. Given two groups that obey universal local limit theorems, it is not known whether their cartesian product also has a universal local limit theorem. We settle the question affirmatively in one case, by considering a random walk on the cartesian product of a nonamenable group whose Cayley graph is a tree, and the integer lattice. As corollaries, we derive large deviations estimates and a central limit theorem.