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 μ that are such that ∑ ρ(|x|)μ(x) < ∞ for increasing regularly varying or slowly varying functions ρ, for instance, s 7→ (1+s)α, α ∈ (0, 2], or s 7→ (1 + log(1 + s))ǫ, ǫ > 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-version of Erschler’s inequality concerning the Følner functions of wreath products. Examples and assorted applications are included.