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 Electronic Journal of Probability > Vol. 10 (2005) > Paper 43 open journal systems 


Random Walks on Groups and Monoids with a Markovian Harmonic Measure

Mairesse Jean, CNRS - Universite Paris 7


Abstract
We consider a transient nearest neighbor random walk on a group G with finite set of generators S. The pair (G,S) is assumed to admit a natural notion of normal form words where only the last letter is modified by multiplication by a generator. The basic examples are the free products of a finitely generated free group and a finite family of finite groups, with natural generators. We prove that the harmonic measure is Markovian of a particular type. The transition matrix is entirely determined by the initial distribution which is itself the unique solution of a finite set of polynomial equations of degree two. This enables to efficiently compute the drift, the entropy, the probability of ever hitting an element, and the minimal positive harmonic functions of the walk. The results extend to monoids.


Full text: PDF

Pages: 1417-1441

Published on: December 16, 2005
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Electronic Journal of Probability. ISSN: 1083-6489