Electronic Journal of Probability

The most visited sites of biased random walks on trees

Yueyun Hu and Zhan Shi

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We consider the slow movement of randomly biased random walk $(X_n)$ on a supercritical 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 Erdős and Révész for simple symmetric random walk on the line.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 62, 14 pp.

Accepted: 10 June 2015
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G50: Sums of independent random variables; random walks 60K37: Processes in random environments

Biased random walk on the Galton--Watson tree branching random walk local time most visited site

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Hu, Yueyun; Shi, Zhan. The most visited sites of biased random walks on trees. Electron. J. Probab. 20 (2015), paper no. 62, 14 pp. doi:10.1214/EJP.v20-4051. https://projecteuclid.org/euclid.ejp/1465067168

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