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

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arxiv 1611.10324 v2 pith:3VZDRYJ5 submitted 2016-11-30 math.PR

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keywords branchingcapacitycriticalrandomwalkdiscretefinitepoint
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abstract

We extend the theory of discrete capacity to critical branching random walk. We introduce branching capacity for any finite subset of $\Z^d, d\geq5$. 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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  1. Yaglom theorem for critical branching random walk on $\mathbb{Z}^d$

    math.PR 2025-12 conditional novelty 7.0 of 10

    Conditioned on hitting a distant set K, the total occupation time of a critical branching random walk is of order ||x||^{4-d} for d≤3, log||x|| for d=4, and bounded for d≥5, with explicit weak limits in all dimensions.

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