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 $\mu$ that are such that $\sum \rho(|x|)\mu(x)<\infty$ for increasing regularly varying or slowly varying functions $\rho$, for instance, $s\mapsto (1+s)^\alpha$, $\alpha\in (0,2]$, or $s\mapsto (1+\log (1+s))^\epsilon$, $\epsilon>0$. 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 F\o lner functions of wreath products. Examples and assorted applications are included.