Maximal displacement in a branching random walk through interfaces

TL;DR

This paper investigates the maximal displacement in a time-inhomogeneous branching random walk, revealing a ballistic growth with a logarithmic correction and bounded fluctuations, based on an environment with piecewise constant reproduction laws.

Contribution

It introduces a new analysis of maximal displacement in time-dependent environments, combining optimization and probabilistic techniques.

Findings

Maximal displacement exhibits ballistic growth with a specific correction.

The correction term is logarithmic in nature.

Fluctuations around the asymptotic behavior are bounded.

Abstract

In this article, we study a branching random walk in an environment which depends on the time. This time-inhomogeneous environment consists of a sequence of macroscopic time intervals, in each of which the law of reproduction remains constant. We prove that the asymptotic behaviour of the maximal displacement in this process consists of a first ballistic order, given by the solution of an optimization problem under constraints, a negative logarithmic correction, plus stochastically bounded fluctuations.