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

TL;DR

This paper investigates the growth rate of maximal and minimal displacements in biased branching random walks on Galton-Watson trees, linking it to large deviation principles of single-particle walks.

Contribution

It provides a characterization of the linear growth speed of extremal displacements using large deviation rate functions, extending understanding of biased branching random walks.

Findings

Maximal displacement grows linearly with time almost surely.

Minimal displacement also exhibits linear growth behavior.

The growth rate is characterized via large deviation rate functions.

Abstract

We study $\lambda$-biased branching random walks on Bienaym\'e--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 $\lambda$-biased random walk of a single particle. A similar result is given for the minimal displacement at time $n$, $\min_{\vert u \vert =n} \vert X(u) \vert$.