Abstract
As we have seen, for every symmetric random walk on a finitely generated group the probability of return to the initial point after 2n steps decays roughly like ϱ 2n, where ϱ ≤ 1 is the so-called spectral radius of the walk. When is it the case that this exponential rate ϱ is strictly less than 1? Kesten [77] discovered that the answer depends on a fundamental structural feature of the ambient group: amenability.
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Lalley, S. (2023). Isoperimetric Inequalities and Amenability. In: Random Walks on Infinite Groups. Graduate Texts in Mathematics, vol 297. Springer, Cham. https://doi.org/10.1007/978-3-031-25632-5_5
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