Random walks in random conductances: Decoupling and spread of infection

Let (G, µ) be a uniformly elliptic random conductance graph on Zd with a Poisson point process of particles at time t = 0 that perform independent simple random walks. We show that inside a cube QK of side length K, if all subcubes of side length ℓ < K inside QK have sufficiently many particles, the particles return to stationarity after cℓ2 time with a probability close to 1. We show that in this setup, an infection spreads with positive speed in any direction. Our framework is robust enough to allow us to also extend the result to infection with recovery.