Biased branching random walks on Bienaym\'e--Galton--Watson trees

• DOI: 10.48550/arxiv.2502.07363
• Publication Date: 2025-01-01
• Authors: Berestycki, Julien; Gantert, Nina; Geldbach, David; Shi, Quan
• Keywords: Probability (math.PR); FOS: Mathematics; 60J80, 60F15, 60F20; Mathematics - Probability

29 pages, 5 figures<br/>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 $n$, $\min_{\vert u \vert =n} \vert X(u) \vert$.<br/>