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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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