Favorite sites of randomly biased walks on a supercritical Galton-Watson tree

Abstract

Erdős and Révész initiated the study of favorite sites by considering the one-dimensional simple random walk. We investigate in this paper the same problem for a class of null-recurrent randomly biased walks on a supercritical Gaton-Watson tree. We prove that there is some parameter $\kappa \in (1,\infty ]$ such that the set of the favorite sites of the biased walk is almost surely bounded in the case $\kappa \in (2,\infty ]$, tight in the case $\kappa =2$, and oscillates between a neighborhood of the root and the boundary of the range in the case $\kappa \in (1,2)$. Moreover, our results yield a complete answer to the cardinality of the set of favorite sites in the case $\kappa \in (2,\infty ]$. The proof relies on the exploration of the Markov property of the local times process with respect to the space variable and on a precise tail estimate on the maximum of local times, using a change of measure for multi-type Galton-Watson trees.

43 pages, 2 figures

References in corpus (1)

• Biased random walks on random graphs