Recurrence of the $ \mathbb{Z}^d$-valued infinite snake via unimodularity

TL;DR

This paper demonstrates that a branching simple random walk on integer lattices is recurrent in dimensions up to four, using unimodular random graph theory to establish the critical dimension for recurrence.

Contribution

It introduces a novel application of unimodular random graph concepts to analyze the recurrence of branching random walks on integer lattices.

Findings

Recurrence occurs if and only if d ≤ 4.

Unimodular random graphs provide a key tool for analyzing recurrence.

The critical dimension for recurrence is identified as four.

Abstract

We use the concept of unimodular random graph to show that the branching simple random walk on $\mathbb{Z}^{d}$ indexed by a critical geometric Galton-Watson tree conditioned to survive is recurrent if and only if $d \leq 4$.