Some examples of random walks on free products of discrete groups

SummaryWe consider the random walk (Xn) associated with a probability p on a free product of discrete groups. Knowledge of the resolvent (or Green's function) of p yields theorems about the asymptotic behaviour of the n-step transition probabilities p*n(x)=P(Xn= x¦ X0=e) as n→∞. Woess [15], Cartwright and Soardi [3] and others have shown that under quite general conditions there is behaviour of the type p*n(x)∼Cxϱ− n n− 3/2. Here we show on the other hand that if G is a free product of m copies ofZr and if (Xn) is the « average » of the classical nearest neighbour random walk on each of the factorsZr, then while it satisfies an « n−3/2 — law » for r small relative to m, it switches to an n− r/2 -law for large r. Using the same techniques, we give examples of irreducible probabilities (of infinite support) on the free groupZ*m which satisfyn−α for $$\lambda \ne \tfrac{3}{2}$$ .