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 $d \leq 4$.
Highlights
Consider a simple random walk on Zd indexed by some tree T
We use the concept of unimodular random graph to show that the branching simple random walk on Zd indexed by a critical geometric Galton-Watson tree conditioned to survive is recurrent if and only if d 4
When T is a critical Galton-Watson tree, the walk is closely related to the theory of superBrownian motion and the associated random snake of Le Gall, see [2, 17]
Summary
Consider a simple random walk on Zd indexed by some tree T. When T is a critical Galton-Watson tree, the walk is closely related to the theory of superBrownian motion and the associated random snake of Le Gall, see [2, 17]. In this note we study the simple random walk on Zd indexed by the critical geometric Galton-Waton tree T∞ conditioned to survive [15]. The simple random walk on Zd indexed by T∞ is recurrent iff d 4. The proof of Theorem 0.1 is based on the use of the “Mass Transport Principle” (see [9, Section 3.2], and [3]) and the related concept of unimodular random graphs combined with simple geometric estimates regarding the tree T∞. We end the note with a few extensions, comments and open questions
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