Random walks on free products of cyclic groups

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-neighbor 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 explicitly solved, to get closed form formulas for the drift. The examples considered are 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.