Tightness of the maximum of branching random walk in random environment and zero-crossings of solutions to discrete parabolic differential equations

TL;DR

This paper proves the tightness of the distribution of the maximum particle displacement in a branching random walk within a random environment, using PDE techniques to analyze zero-crossings of solutions to discrete parabolic equations.

Contribution

It extends previous tightness results from annealed to quenched settings without extra environment assumptions, employing PDE analysis for the first time in this context.

Findings

Distribution of maximum displacement is tight for almost every environment realization.

Zero-crossings of solutions to related PDEs are non-increasing over time.

Method applies without additional assumptions on the environment.

Abstract

We study branching random walk on $\mathbb{Z}$ in a bounded i.i.d. random environment. For this process, we prove that, for almost every realization of the environment, the distributions of the maximally displaced particle (re-centered around their medians) are tight. This extends the result of arXiv:2408.01555 , where tightness was established in the annealed sense, and of arXiv:2212.12390 , where a similar quenched result was proved for branching Brownian motion in random environment. Our proof relies on studying certain discrete-space linear PDEs and establishing that the number of zero-crossings of their solutions is non-increasing in time. We observe that our technique requires no additional assumptions on the environment, in contrast to arXiv:2408.01555 , arXiv:2212.12390 .