### Abstract

^{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 |
---|---|

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 |

### Fingerprint

### Keywords

- math.PR

### Cite this

**Multi-scale Lipschitz percolation of increasing events for Poisson random walks.** / Gracar, Peter; Stauffer, Alexandre.

Research output: Contribution to journal › Article

*Annals of Applied Probability*, vol. 29, no. 1, pp. 376-433. https://doi.org/10.1214/18-AAP1420

}

TY - JOUR

T1 - Multi-scale Lipschitz percolation of increasing events for Poisson random walks

AU - Gracar, Peter

AU - Stauffer, Alexandre

N1 - The text has been improved throughout and the proofs clarified

PY - 2019/2/28

Y1 - 2019/2/28

N2 - Consider the graph induced by Zd, equipped with uniformly elliptic random conductances. At time 0, place a Poisson point process of particles on Zd and let them perform independent simple random walks. Tessellate the graph into cubes indexed by i ∈ Zd 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.

AB - Consider the graph induced by Zd, equipped with uniformly elliptic random conductances. At time 0, place a Poisson point process of particles on Zd and let them perform independent simple random walks. Tessellate the graph into cubes indexed by i ∈ Zd 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.

KW - math.PR

U2 - 10.1214/18-AAP1420

DO - 10.1214/18-AAP1420

M3 - Article

VL - 29

SP - 376

EP - 433

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 1

ER -