## The Annals of Applied Probability

### Multi-scale Lipschitz percolation of increasing events for Poisson random walks

#### 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.

#### Article information

Source
Ann. Appl. Probab., Volume 29, Number 1 (2019), 376-433.

Dates
Revised: May 2018
First available in Project Euclid: 5 December 2018

https://projecteuclid.org/euclid.aoap/1544000432

Digital Object Identifier
doi:10.1214/18-AAP1420

Mathematical Reviews number (MathSciNet)
MR3910007

Zentralblatt MATH identifier
07039128

#### Citation

Gracar, Peter; Stauffer, Alexandre. Multi-scale Lipschitz percolation of increasing events for Poisson random walks. Ann. Appl. Probab. 29 (2019), no. 1, 376--433. doi:10.1214/18-AAP1420. https://projecteuclid.org/euclid.aoap/1544000432

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