Connectivity properties of Branching Interlacements

We consider connectivity properties of the Branching Interlacements model in Z^d, d ≥ 5, recently introduced by Angel, Ráth and Zhu (Angel et al., 2019). Using stochastic dimension techniques we show that every two vertices visited by the branching interlacements are connected via at most [d/4] conditioned critical branching random walks from the underlying Poisson process, and that this upper bound is sharp. In particular every such two branching random walks intersect if and only if 5 ≤ d ≤ 8. The stochastic dimension of branching random walk result is of independent interest. We additionally obtain heat kernel bounds for branching random walks conditioned on survival.