Summary
We 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 ofZ r and if (Xn) is the « average » of the classical nearest neighbour random walk on each of the factorsZ r, 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\(\lambda \ne \tfrac{3}{2}\).
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Cartwright, D.I. Some examples of random walks on free products of discrete groups. Annali di Matematica pura ed applicata 151, 1–15 (1988). https://doi.org/10.1007/BF01762785
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DOI: https://doi.org/10.1007/BF01762785