Intersections of branching random walks on $\mathbb{Z}^8$

Abstract

We consider random walks on $\Z^8$ indexed by the infinite invariant tree, which consists of an infinite spine and finite random trees attached to it on both sides. We establish the precise order of the non-intersection probability between one walk indexed by one side of the tree, and an independent one indexed by both sides of an independent tree. This is analogous to the result by Lawler from the '90s for two independent simple random walks on $\Z^4$. We also prove a weak law of large numbers for the branching capacity of the range of a branching random walk.