Branching interlacements and tree-indexed random walks in tori

Abstract

We introduce a model of branching interlacements for general critical offspring distributions. It consists of a countable collection of infinite tree-indexed random walk trajectories on $Z^d,d\geq5$. We show that this model turns out to be the local limit of the tree-indexed random walk in a discrete torus, conditioned on the size proportional to the volume of the torus. This generalizes the previous results of Angel, Ráth and the author, for the critical geometric offspring distribution. Our model also includes the model of random interlacements introduced by Sznitman as a degenerate case. To obtain the local convergence, we establish results on decomposing large random trees into small trees, local limits of random trees around a prefixed vertex, and asymptotics of the visiting probability of a set by a tree-indexed random walk with a given size in a torus. These auxiliary results are interesting in themselves. As another application, we show that when $d\geq5$ the cover time of a $d$-dimensional torus of side-length $N$ by tree-indexed random walks is concentrated at $N^d log N^d/\mathrm{BCap}(\{0\})$.