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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.

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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