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

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 $κ\in (1, \infty]$ such that the set of the favorite sites of the biased walk is almost surely bounded in the case $κ\in (2, \infty]$, tight in the case $κ=2$, and oscillates between a neighborhood of the root and the boundary of the range in the case $κ\in (1, 2)$. Moreover, our results yield a complete answer to the cardinality of the set of favorite sites in the case $κ\in (2, \infty]$. 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.

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