Low-dimensional lonely branching random walks die out

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.