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

TL;DR

This paper investigates the maximal displacement in a supercritical branching random walk within a complex, two-level random environment, revealing a new asymptotic behavior influenced by environmental randomness.

Contribution

It introduces a model with two levels of randomness and determines the asymptotic maximal displacement, showing it exceeds the usual logarithmic correction seen in homogeneous cases.

Findings

Maximal displacement follows $V_n - heta \log n + o_P(\log n)$ with a random walk-like $V_n$

The correction term $ heta$ is larger than in homogeneous branching random walks

Environmental randomness significantly impacts the displacement's asymptotic behavior

Abstract

The behavior of the maximal displacement of a supercritical branching random walk has been a subject of intense studies for a long time. But only recently the case of time-inhomogeneous branching has gained focus. The contribution of this paper is to analyze a time-inhomogeneous model with two levels of randomness. In the first step a sequence of branching laws is sampled independently according to a distribution on the set of point measures' laws. Conditionally on the realization of this sequence (called environment) we define a branching random walk and find the asymptotic behavior of its maximal particle. It is of the form $V_n -\varphi \log n + o_\mathbf{P}(\log n)$, where $V_n$ is a function of the environment that behaves as a random walk and $\varphi>0$ is a deterministic constant, which turns out to be bigger than the usual logarithmic correction of the homogeneous branching random walk.