On the critical branching random walk I: Branching capacity and visiting probability

Abstract

We extend the theory of discrete capacity to critical branching random walk. We introduce branching capacity for any finite subset of ${\mathbb{Z}}^d,d\ge 5$. Analogous to the regular discrete capacity, branching capacity is closely related to the asymptotics of the probability of visiting a fixed finite set by a critical branching random walk starting from a distant point and the conditional distribution of the hitting point.

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