Coexistence for Competing Branching Random Walks with Identical Asymptotic Shape on $\mathbb{Z}^d$

TL;DR

This paper investigates the conditions under which two independent branching random walks on a lattice can coexist with infinitely many vertices of both colors, analyzing their asymptotic shapes and extremal particles.

Contribution

It establishes coexistence criteria for branching random walks with identical asymptotic shapes and provides examples of almost-sure non-coexistence in one dimension.

Findings

Coexistence occurs under mild conditions for walks with similar extremal behavior.

In one dimension, coexistence can be almost surely absent when asymptotic shapes match.

Second-order and large deviation techniques are key tools in the analysis.

Abstract

We consider two independent branching random walks that start next to each other on the $d$-dimensional hypercubic lattice and that carry two different colors. Vertices of the lattice are colored according to the color of the walker cloud that first visits the vertex, leading to the question of possible coexistence in the sense that both colors appear on infinitely many vertices. Under mild conditions, we prove the coexistence for two independently distributed branching random walks obeying the same first- and second-order behavior for their extremal particles. To complement this result, we also exhibit examples for the almost-sure absence of coexistence, for $d=1$, in cases where the asymptotic shapes of the walker clouds are calibrated to coincide, thereby answering a question by Deijfen and Vilkas (ECP 28(15):1-11, 2023). As a main tool we employ second-order and large-deviation approximations for the position of the extremal particles in one-dimensional branching random walks.