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 probabilityp(n,x,y) that a particle starting atxreachesyat timenasngoes to infinity is established.