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.

Additional explanations and figures. arXiv admin note: text overlap with arXiv:1108.6322

References in corpus (4)

• Random walks on supercritical percolation clusters
• The spread of a rumor or infection in a moving population
• Random walks in random conductances: decoupling and spread of infection
• Absorbing-state transition for Stochastic Sandpiles and Activated Random Walks

Cited by in corpus (2)

• Local survival of spread of infection among biased random walks
• Local and global survival for infections with recovery