Random walks and isoperimetric profiles under moment conditions

Let $G$ be a finitely generated group equipped with a finite symmetric generating set and the associated word length function $|\cdot |$. We study the behavior of the probability of return for random walks driven by symmetric measures $μ$ that are such that $\sum ρ(|x|)μ(x)<\infty$ for increasing regularly varying or slowly varying functions $ρ$, for instance, $s\mapsto (1+s)^α$, $α\in (0,2]$, or $s\mapsto (1+\log (1+s))^ε$, $ε>0$. For this purpose we develop new relations between the isoperimetric profiles associated with different symmetric probability measures. These techniques allow us to obtain a sharp $L^2$-version of Erschler's inequality concerning the Følner functions of wreath products. Examples and assorted applications are included.

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