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

Peter Gracar, Alexandre Stauffer

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Abstract

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.
Original languageEnglish
Pages (from-to)376-433
Number of pages58
JournalAnnals of Applied Probability
Volume29
Issue number1
DOIs
Publication statusPublished - 1 Feb 2019

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Lipschitz
Random walk
Siméon Denis Poisson
Regular hexahedron
Poisson Point Process
Simple Random Walk
Interval
Graph in graph theory
Conductance
Infection
Infinity
Cell
Graph

Keywords

  • math.PR

Cite this

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

In: Annals of Applied Probability, Vol. 29, No. 1, 01.02.2019, p. 376-433.

Research output: Contribution to journalArticle

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