Random walks in random conductances: decoupling and spread of infection

Let $(G,\mu)$ be a uniformly elliptic random conductance graph on $\mathbb{Z}^d$ with a Poisson point process of particles at time $t=0$ that perform independent simple random walks. We show that inside a cube $Q_K$ of side length $K$, if all subcubes of side length $\ell<K$ inside $Q_K$ have sufficiently many particles, the particles return to stationarity after $c\ell^2$ time with a probability close to $1$. We also show this result for percolation clusters on locally finite graphs. Using this mixing result, 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, where we show positive speed and that the infection survives indefinitely with positive probability.