### Abstract

Let (G,μ) be a uniformly elliptic random conductance graph on Z
^{d} with a Poisson point process of particles at time t=0 that perform independent simple random walks. We show that inside a cube Q
_{K} of side length K, if all subcubes of side length ℓ<K inside Q
_{K} have sufficiently many particles, the particles return to stationarity after cℓ
^{2} time with a probability close to 1. We show that in this setup, an infection spreads with positive speed in any direction. Our framework is robust enough to allow us to also extend the result to infection with recovery.

Original language | English |
---|---|

Pages (from-to) | 3547-3569 |

Number of pages | 23 |

Journal | Stochastic Processes and their Applications |

Volume | 129 |

Issue number | 9 |

Early online date | 5 Oct 2018 |

DOIs | |

Publication status | Published - 1 Sep 2019 |

### Fingerprint

### Keywords

- math.PR

### Cite this

**Random walks in random conductances : decoupling and spread of infection.** / Gracar, Peter; Stauffer, Alexandre.

Research output: Contribution to journal › Article

*Stochastic Processes and their Applications*, vol. 129, no. 9, pp. 3547-3569. https://doi.org/10.1016/j.spa.2018.09.016

}

TY - JOUR

T1 - Random walks in random conductances

T2 - decoupling and spread of infection

AU - Gracar, Peter

AU - Stauffer, Alexandre

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

PY - 2019/9/1

Y1 - 2019/9/1

N2 - Let (G,μ) be a uniformly elliptic random conductance graph on Z d with a Poisson point process of particles at time t=0 that perform independent simple random walks. We show that inside a cube Q K of side length K, if all subcubes of side length ℓ<K inside Q K have sufficiently many particles, the particles return to stationarity after cℓ 2 time with a probability close to 1. We show that in this setup, an infection spreads with positive speed in any direction. Our framework is robust enough to allow us to also extend the result to infection with recovery.

AB - Let (G,μ) be a uniformly elliptic random conductance graph on Z d with a Poisson point process of particles at time t=0 that perform independent simple random walks. We show that inside a cube Q K of side length K, if all subcubes of side length ℓ<K inside Q K have sufficiently many particles, the particles return to stationarity after cℓ 2 time with a probability close to 1. We show that in this setup, an infection spreads with positive speed in any direction. Our framework is robust enough to allow us to also extend the result to infection with recovery.

KW - math.PR

U2 - 10.1016/j.spa.2018.09.016

DO - 10.1016/j.spa.2018.09.016

M3 - Article

VL - 129

SP - 3547

EP - 3569

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 9

ER -