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### Abstract

Consider the graph induced by Z

^{d}, equipped with uniformly elliptic random conductances. At time 0, place a Poisson point process of particles on Z^{d}and let them perform independent simple random walks. Tessellate the graph into cubes indexed by*i*∈ Z^{d}and tessellate time into intervals indexed by τ. Given a local event*E*(*i,τ*) that depends only on the particles inside the space time region given by the cube*i*and the time interval*τ*, we prove the existence of a Lipschitz connected surface of cells (*i,τ*) that separates the origin from inﬁnity on which*E*(*i, τ*) 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.Original language | English |
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Pages (from-to) | 376-433 |

Number of pages | 58 |

Journal | Annals of Applied Probability |

Volume | 29 |

Issue number | 1 |

Early online date | 5 Dec 2018 |

DOIs | |

Publication status | Published - 28 Feb 2019 |

### Keywords

- math.PR

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## Projects

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