2016/11/14 by Chen, Dayue, de Raphélis, Loïc, Hu, Yueyun
#60G50 #60J80 #60K37 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1611.04497
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∈ (1, ∞] such that the set of the favorite sites of the biased walk is almost surely bounded in the case κ∈ (2, ∞], tight in the case κ=2, and oscillates between a neighborhood of the root and the boundary of the range in the case κ∈ (1, 2). Moreover, our results yield a complete answer to the cardinality of the set of favorite sites in the case κ∈ (2, ∞]. 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.