Resumen de Random walks and isoperimetric profiles under moment conditions

Laurent Saloff-Coste, Tianyi Zheng

• Let GG 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 μμ that are such that ∑ρ(|x|)μ(x)<∞∑ρ(|x|)μ(x)<∞ for increasing regularly varying or slowly varying functions ρρ, for instance, s↦(1+s)αs↦(1+s)α, α∈(0,2]α∈(0,2], or s↦(1+log(1+s))εs↦(1+log⁡(1+s))ε, ε>0ε>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 L2L2-version of Erschler’s inequality concerning the Følner functions of wreath products. Examples and assorted applications are included.