Some examples of random walks on free products of discrete groups (Q1113161)

Let G be the free product of a family of discrete groups \(G_ j\), \(j\in J\), and let p be a probability measure on G, which is a convex combination of probabilities \(p_ j\) on \(G_ j\). The asymptotic behaviour of convolution products \(p^{*n}\) is investigated. It is known [see e.g. \textit{W. Woess}, Boll. Unione Mat. Ital., VI. Ser., B5 961-982 (1986; Zbl 0627.60012)] that under certain restrictions we have the behaviour \[ p^{*n}(x)\sim C(x)\rho^{-n} n^{-3/2}, \] \(\rho\) being the radius of convergence of the power series \(\sum p^{*n}(x)z^ n\). First the author reproves some of the results of Woess, showing that also \(n^{-\lambda}\)-laws (\(\lambda\neq 3/2)\) occur. Such examples are explicitely constructed: products of m copies of nearest neighbour random walks on \({\mathbb{Z}}^ r\), r-sufficiently large, or of symmetric random walks on free products of \({\mathbb{Z}}\), if the measures \(p_ j\) are not finitely supported.