Tightness for branching random walk in a space-inhomogeneous random environment

TL;DR

This paper studies the maximum displacement of a branching random walk in a space-inhomogeneous random environment on integers, proving tightness of the centered maximum in an annealed sense.

Contribution

It establishes the tightness of the maximum of the branching random walk in a randomized, inhomogeneous environment, with a novel environment-dependent centering.

Findings

Maximum $M_t$ is tight after centering by $ ilde{m}_t$

Centering $ ilde{m}_t$ depends only on the environment

Results hold in an annealed probabilistic sense

Abstract

We consider the maximum $M_t$ of branching random walk in a space-inhomogeneous random environment on $\mathbb{Z}$. In this model the branching rate while at some location $x\in\mathbb{Z}$ is randomized in an i.i.d. manner. We prove that there is a centering $\widetilde{m}_t$ depending only on the environment such that $(M_t-\widetilde{m}_t)_{t\ge 0}$ is tight in an annealed sense.