Explicit bounds for the return probability of simple random walks

摘要

We give an exact computation of the second order term in the asymptotic expansion of the return probability, P-2n(d) (0,0), of a simple random walk on the d-dimensional cubic lattice. We also give an explicit bound on the remainder. In particular, we show that P-2n(d) (0,0) 2(d/4 pi n)(d/2) where n >= M = M(d) is explicitly given.