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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.

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