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

Consider the graph induced by .d, equipped with uniformly elliptic random conductances.At time 0, place a Poisson point process of particles on .d and let them perform independent simple random walks.Tessellate the graph into cubes indexed by i..d and tessellate time into intervals indexed by ..Given a local event E(i,.) that depends only on the particles inside the space time region given by the cube i and the time interval ., we prove the existence of a Lipschitz connected surface of cells (i,.) that separates the origin from infinity on which E(i,.) 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.