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

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.