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On drift and entropy growth for random walks on groups. (English) Zbl 1043.60005

Let \(\mu\) be a probability measure with finite support on a discrete group \(G\) such that the support of \(\mu\) generates \(G\). The author investigates the asymptotic behavior of the entropy \[ H(n)=- \sum_{g\in G}\mu^{(n)}(g)\ln \mu^{(n)}(g) \] and the drift \(L(n):= \int_Gl(g)\,d\mu^{(n)}(g)\) for the length function \(l\) on \(G\) associated with the support of \(\mu\) as generatig set. In particular, inequalities relating \(H\), \(L\), and the growth of \(G\) are derived. As an application, examples of random walks with \(L(n)\simeq n/\ln^{(k)}n\) for \(n\to\infty\) and any \(k\) are constructed, where \(\ln^{(k)}\) denotes the \(k\)th iterate of the logarithm.

MSC:

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
43A05 Measures on groups and semigroups, etc.
43A07 Means on groups, semigroups, etc.; amenable groups
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References:

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