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J i on Electr o u a rn l o f P c r ob abil ity Vol 10 2005 Paper no 43 pages 1417 1441 Journal URL http www math washington edu ejpecp Random Walks on Groups and Monoids with a Markovian Harmonic Measure Jean Mairesse LIAFA CNRS Universite Paris 7 case 7014 2 place Jussieu 75251 Paris Cedex 05 France E mail Jean Mairesse liafa jussieu fr Abstract We consider a transient nearest neighbor random walk on a group G with finite set of generators The pair G 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 Keywords Finitely generated group or monoid free product random walk harmonic measure AMS classification 2000 Primary 60J10 60B15 31C05 Secondary 60J22 65C40 20F65 Submitted to EJP on November 9 2005 Final version accepted on December 1 2005 1417 1 Introduction Consider a Markov chain whose state space is the Cayley graph of a group G with a finite set of generators If the Markov chain moves from a node g to a node g h with a probability h depending only on h we say that it is a randow walk If the support of is included in we say that we have a nearest neighbor random walk with respect to A remarkable and well known result is that the harmonic measure of a transient nearest neighbor random walk on a finitely generated free group is Markovian 6 19 25 This can be viewed as a consequence of the multiplicative structure of the Green kernel If one



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