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, R\'{a}th 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 $N^d log N^d/\mathrm{BCap}(\{0\})$.