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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
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