The most visited sites of biased random walks on trees

Abstract : We consider the slow movement of randomly biased random walk (Xn) on a supercriti-cal Galton–Watson tree, and are interested in the sites on the tree that are most visited by the biased random walk. Our main result implies tightness of the distributions of the most visited sites under the annealed measure. This is in contrast with the one-dimensional case, and provides, to the best of our knowledge, the first non-trivial example of null recurrent random walk whose most visited sites are not transient, a question originally raised by Erd˝ os and Révész [11] for simple symmetric random walk on the line.
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Yueyun Hu, Zhan Shi. The most visited sites of biased random walks on trees. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2015, 20, pp.1-14. ⟨10.1214/EJP.v20-4051⟩. ⟨hal-01214677⟩

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