Some examples of random walks on free products of discrete groups |
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Authors: | Donald I Cartwright |
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Institution: | (1) Present address: Department of Pure Mathematics, The University of Sydney, 2006 Sydney, New South Wales, Australia |
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Abstract: | 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)Cx– 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
– for
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