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Multi-scale Lipschitz percolation of increasing events for Poisson random walks. (English) Zbl 1408.60038
Summary: Consider the graph induced by $$\mathbb{Z}^d$$, equipped with uniformly elliptic random conductances. At time 0, place a Poisson point process of particles on $$\mathbb{Z}^d$$ and let them perform independent simple random walks. Tessellate the graph into cubes indexed by $$i\in\mathbb{Z}^d$$ and tessellate time into intervals indexed by $$\tau$$. Given a local event $$E(i,\tau)$$ that depends only on the particles inside the space time region given by the cube $$i$$ and the time interval $$\tau$$, we prove the existence of a Lipschitz connected surface of cells $$(i,\tau)$$ that separates the origin from infinity on which $$E(i,\tau)$$ holds. This gives a directly applicable and robust framework for proving results in this setting that need a multi-scale argument. For example, this allows us to prove that an infection spreads with positive speed among the particles.

MSC:
 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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