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Random walks and isoperimetric profiles under moment conditions. (English) Zbl 1378.60019
Let $$G$$ be a finitely generated group equipped with a finite symmetric generating set and the associated word length function $$\left| \cdot \right|$$. The main aim of this paper is to study the behaviour of the probability of return for a random walk driven by a symmetric measure $$\mu$$ such that $$\sum \rho (\left| x\right| )\mu (x)<\infty$$ for a slowly varying function $$\rho$$. The authors present new relations between the isoperometric profiles associated with different symmetric probability measures and a sharp $$L^{2}$$-version of Erschler’s inequality concerning Følner functions of wreath products. Some applications of the obtained results are also presented.

##### MSC:
 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 60B05 Probability measures on topological spaces 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60G50 Sums of independent random variables; random walks 20F65 Geometric group theory
##### Keywords:
random walk; isoperimetric profile; wreath product; moment
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