Asymptotic behavior of the transition probability of a random walk on an infinite graph

Ideas cultivated in spectral geometry are applied to obtain an asymptotic property of a reversible random walk on an infinite graph satisfying a certain periodic condition. In the course of our argument, we employ perturbation theory for the maximal eigenvalues of twisted transition operator. As a result, an asymptotic of the probability p(n, x, y) that a particle starting at x reaches y at time n as n goes to infinity is established. (C) 1998 Academic Press.