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

Benjamini, Itai; Curien, Nicolas

Electron. Commun. Probab. 17, Paper No. 1, 10 p. (2012).

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)

Keywords:

Galton-Watson trees; random snake; mass-transport; unimodular random graph