Recurrence of the $ \mathbb{Z}^d$-valued infinite snake via   unimodularity
  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 $d \leq 4$. 
Itai Benjamini and Nicolas Curien
math.PR