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.