Random walks on free products of cyclic groups

Details Random walks on free products of cyclic groups Random walks on free products of cyclic groups

2005-09-09

arxiv.org/abs/math/0509211v3

Mathematics

Probability

20 pages, 3 figures. Added a paragraph on Free products of countable groups. Added an example in the Extremal generators secti

Scientific paper

Let G be a free product of a finite family of finite groups, with the set of generators being formed by the union of the finite groups. We consider a transient nearest-neighbour random walk on G. We give a new proof of the fact that the harmonic measure is a special Markovian measure entirely determined by a finite set of polynomial equations. We show that in several simple cases of interest, the polynomial equations can be explicitely solved, to get closed form formulas for the drift. The examples considered are the modular group Z/2Z*Z/3Z, Z/3Z*Z/3Z, Z/kZ*Z/kZ, and the Hecke groups Z/2Z*Z/kZ. We also use these various examples to study Vershik's notion of extremal generators, which is based on the relation between the drift, the entropy, and the volume of the group.

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