Some examples of random walks on free products of discrete groups (original) (raw)

We consider the random walk (X~) associated with a probability p on a #ee produet o] discrete groups. Knowledge o] the resolvent (or Green's ]unction) o] p yields theorems about the asymptotic behaviour o] the n-step transition probabilities p*~(x) = P(X n = x I X o = e) as n-> c~. Woess [15], Cartwright and Soardi [3] and others have shown that under quite general conditions there is behaviour o] the type p*~(x)~ C~p-~n-~. Here we show on the other hand that i]G is a #so product o] ~ copies o] Z r and i] (Xn) is the ((average ~ o] the classical nearest neighbour random walk on each o] the ]actors Z% then while it satis]ies an (~ n-~-law ~) ]or r small relative to m, it switches to an n-r/2-1aw ]or large r. Using the same techniques, we give examples o] irreducible probabilities (o] in]inite support) on the #ee group Z *~ which satis]y n-~-laws ]or 2 #-~- .