Maximal displacement of a branching random walk in time-inhomogeneous environment

TL;DR

This paper analyzes the maximal displacement of a branching random walk in a time-inhomogeneous environment, providing precise asymptotics and partially confirming a conjecture, with potential applications to related quantities.

Contribution

It computes the first two terms of the asymptotic maximal displacement in a time-inhomogeneous setting, introducing new techniques for such environments.

Findings

First-order term obtained from an optimization problem.

Second-order term of order n^{1/3} from random walk estimates.

Partial confirmation of Fang and Zeitouni's conjecture.

Abstract

Consider a branching random walk evolving in a macroscopic time-inhomogeneous environment, that scales with the length $n$ of the process under study. We compute the first two terms of the asymptotic of the maximal displacement at time $n$. The coefficient of the first (ballistic) order is obtained as the solution of an optimization problem, while the second term, of order $n^{1/3}$, comes from time-inhomogeneous random walk estimates, that may be of independent interest. This result partially answers a conjecture of Fang and Zeitouni. Same techniques are used to obtain the asymptotic of other quantities, such as the consistent maximal displacement.