## Recurrence of the $$\mathbb{Z}^d$$-valued infinite snake via unimodularity.(English)Zbl 1244.60085

Summary: 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 $$d \leq 4$$.

### MSC:

 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 05C80 Random graphs (graph-theoretic aspects)
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