On some random walks driven by spread-out measures

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

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