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

Abstract

We study $\lambda$-biased branching random walks on Bienaymé--Galton--Watson trees in discrete time. We consider the maximal displacement at time $n$, ${\max }_{|u|=n}|X(u)|$, 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 $\lambda$-biased random walk of a single particle. A similar result is given for the minimal displacement at time $n$, ${\min }_{|u|=n}|X(u)|$.

29 pages, 4 figures -- to appear in AIHP