Analytic Combinatorics
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Suppose we are given finitely generated groups Γ1,…,Γm equipped with irreducible random walks. Thereby we assume that the expansions of the corresponding Green functions at their radii of convergence contain only logarithmic or algebraic terms as singular terms up to sufficiently large order (except for some degenerate cases). We consider transient random walks on the free product Γ1* … *Γm and give a complete classification of the possible asymptotic behaviour of the corresponding n-step return probabilities. They either inherit a law of the form ϱnδn**math-image** log **math-image**n from one of the free factors Γi or obey a ϱnδn−3/2-law, where ϱ \documentclass{article} \usepackage{amsmath, amsthm, amssymb, amsfonts}\pagestyle{empty}\begin{document} $\mathbb{Z}^{d_1}\ast \ldots \ast \mathbb{Z}^{d_m}$ \end{document} **image** . Moreover, we characterize the possible phase transitions of the non-exponential types n**math-image** log **math-image**n in the case Γ1 * Γ2. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012 © 2012 Wiley Periodicals, Inc.