Return Probabilities of Random Walks on Non- Amenable Groups
A fundamental theorem of Harry Kesten asserts that the return probabilities of a random walk on a non-amenable group must always decay exponentially with the number of steps. Is there always a sharp asymptotic formula for the return probabilities, analogous to the Local Limit Theorem for random walks on Zn? If so, is the nature of the formula (in particular, the polynomial correction to the exponential factor) determined by the geometry of the group? We will review recent progress on these questions.