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February 2019 Multi-scale Lipschitz percolation of increasing events for Poisson random walks
Peter Gracar, Alexandre Stauffer
Ann. Appl. Probab. 29(1): 376-433 (February 2019). DOI: 10.1214/18-AAP1420

Abstract

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.

Citation

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Peter Gracar. Alexandre Stauffer. "Multi-scale Lipschitz percolation of increasing events for Poisson random walks." Ann. Appl. Probab. 29 (1) 376 - 433, February 2019. https://doi.org/10.1214/18-AAP1420

Information

Received: 1 July 2017; Revised: 1 May 2018; Published: February 2019
First available in Project Euclid: 5 December 2018

zbMATH: 07039128
MathSciNet: MR3910007
Digital Object Identifier: 10.1214/18-AAP1420

Subjects:
Primary: 60G55
Secondary: 60K35

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 1 • February 2019
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