Low-dimensional lonely branching random walks die out

TL;DR

This paper proves that low-dimensional lonely branching random walks on integer lattices die out locally if the underlying walk is recurrent, even with additional branching when not alone.

Contribution

It establishes the local extinction of lonely branching random walks in low dimensions under recurrence, extending previous results to include extra branching mechanisms.

Findings

Lonely branching random walks die out locally in recurrent low dimensions.

Additional branching when not alone does not prevent local extinction.

Results apply to symmetric recurrent random walks on Z^d.

Abstract

The lonely branching random walks on ${\mathbb Z}^d$ is an interacting particle system where each particle moves as an independent random walk and undergoes critical binary branching when it is alone. We show that if the symmetrized walk is recurrent, lonely branching random walks die out locally. Furthermore, the same result holds if additional branching is allowed when the walk is not alone.