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 $τ$. 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.