Subsequential tightness for branching random walk in random environment

TL;DR

This paper proves the existence of deterministic subsequences along which the maximum of branching random walk in random environment remains tightly bounded, advancing understanding of its extremal behavior.

Contribution

It establishes subsequential tightness for BRWRE maxima, partially answering an open question and adapting methods from bounded increment cases.

Findings

Existence of subsequences with tight maxima in BRWRE

Partial resolution of an open problem in the field

Method adaptation from bounded to random environments

Abstract

We consider branching random walk in random environment (BRWRE) and prove the existence of deterministic subsequences along which their maximum, centered at its mean, is tight. This partially answers an open question in arXiv:1711.00852. The method of proof adapts an argument developed by Dekking and Host for branching random walks with bounded increments. The question of tightness without the need for subsequences remains open.