Random walks and isoperimetric profiles under moment conditions

• Laurent Saloff-Coste ^[1] ; Tianyi Zheng ^[2]
  1. [1] Cornell University

     Cornell University

     City of Ithaca, Estados Unidos

     
  2. [2] Stanford University

     Stanford University

     Estados Unidos

     
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 6, 2016, págs. 4133-4183
• Idioma: inglés
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• Resumen

  • 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.