On some random walks driven by spread-out measures

Abstract

Let $G$ be a finitely generated group equipped with a symmetric generating -tuple $S$. Let $|\cdot |$ and $V$ be the associated word length and volume growth function. Let $\nu$ be a probability measure such that . We prove that if $G$ has polynomial volume growth then ${\nu }^{(n)}(e)\simeq V(\sqrt{n\log n}{)}^{-1}$. We also obtain assorted estimates for other spread-out probability measures.