Biased branching random walks on Bienaymé--Galton--Watson trees

2025/02/11 by Berestycki, Julien, Gantert, Nina, Geldbach, David +1
#60F15 #60F20 #60J80 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2502.07363

Abstract

We study λ-biased branching random walks on Bienaymé--Galton--Watson trees in discrete time. We consider the maximal displacement at time n, max_{\vert u \vert =n} \vert X(u) \vert, and show that it almost surely grows at a deterministic, linear speed. We characterize this speed with the help of the large deviation rate function of the λ-biased random walk of a single particle. A similar result is given for the minimal displacement at time n, min_{\vert u \vert =n} \vert X(u) \vert.

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