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Random Walks on Trees with Finitely Many Cone Types
Authors:Tatiana Nagnibeda  Wolfgang Woess
Institution:(1) Department of Mathematics, Royal Institute of Technology, S 100 44 Stockholm, Sweden;(2) Institut für Mathematik, Technische Universität Graz, Steyrergasse 30, A-8010 Graz, Austria
Abstract:This paper is devoted to the study of random walks on infinite trees with finitely many cone types (also called periodic trees). We consider nearest neighbour random walks with probabilities adapted to the cone structure of the tree, which include in particular the well studied classes of simple and homesick random walks. We give a simple criterion for transience or recurrence of the random walk and prove that the spectral radius is equal to 1 if and only if the random walk is recurrent. Furthermore, we study the asymptotic behaviour of return probabilitites and prove a local limit theorem. In the transient case, we also prove a law of large numbers and compute the rate of escape of the random walk to infinity, as well as prove a central limit theorem. Finally, we describe the structure of the boundary process and explain its connection with the random walk.
Keywords:tree  random walk  transience  rate of escape
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