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 |.|. We study the behavior of the probability of return for random walks driven by symmetric measures . that are such that ..(|x|).(x)<. for increasing regularly varying or slowly varying functions ., for instance, s.(1+s)., ..(0,2], or s.(1+log(1+s))., .>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 L2-version of Erschler.s inequality concerning the Følner functions of wreath products. Examples and assorted applications are included.