Random walks on groups with a tree-like Cayley graphs

This paper announces results which have been later developped in three articles: 1. "Random walks on groups and monoids with a Markovian harmonic measure", Electron. J. Probab., vol. 10, p. 1417-1441, 2005. 2. "Random walks on free products of cyclic groups", J. London Math. Society, vol. 75, n. 1, p. 47-66, 2007. 3. "Randomly growing braid on three strands and the manta ray", Ann. Appl. Probab., vol. 17, n. 2, p. 502-536, 2007.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 which are modified only locally when multiplied by generators. 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 and can be completely described via a finite set of polynomial equations. It enables to 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. In several simple cases of interest, the set of polynomial equations can be explicitly solved, to get closed form formulas for the drift, the entropy,... Various examples are treated: the modular group Z/2Z *Z/3Z, the Hecke groups Z/2Z *Z/kZ, the free products of two isomorphic cyclic groups Z/kZ *Z/kZ, the braid group B3 , and Artin groups with two generators