MULTI-SCALE LIPSCHITZ PERCOLATION OF INCREASING EVENTS FOR POISSON RANDOM WALKS

Consider the graph induced by ℤ d , equipped with uniformly elliptic random conductances. At time 0, place a Poisson point process of particles on ℤ d and let them perform independent simple random walks. Tessellate the graph into cubes indexed by i ∈ 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.