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UBC Theses and Dissertations

Critical branching random walks, branching capacity and branching interlacements Zhu, Qingsan


This thesis concerns critical branching random walks. We focus on supercritical (d ≥ 5 or higher) and critical (d=4) dimensions. In this thesis, we extend the potential theory for random walk to critical branching random walk. In supercritical dimensions, we introduce branching capacity for every finite subset of ℤ^d and construct its connections with critical branching random walk through the following three perspectives. 1. The visiting probability of a finite set by a critical branching random walk starting far away; 2. Branching recurrence and branching transience; 3. Local limit of branching random walk in torus conditioned on the total size. Moreover, we establish the model which we call 'branching interlacements' as the local limit of branching random walk in torus conditioned on the total size. In the critical dimension, we also construct some parallel results. On the one hand, we give the asymptotics of visiting a finite set and the convergence of the conditional hitting point. On the other hand, we establish the asymptotics of the range of a branching random walk conditioned on the total size. Also in this thesis, we analyze a small game which we call the Majority-Markov game and give an optimal strategy.

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