# Random walks in random conductances: decoupling and spread of infection

## TL;DR

This paper demonstrates that particles in random conductance graphs and percolation clusters rapidly reach stationarity within subregions, enabling the analysis of infection spread with positive speed and indefinite survival, even with recovery.

## Contribution

It introduces a novel mixing time result for particles in random conductance graphs and applies it to analyze infection spread and survival in these environments.

## Key findings

- Particles return to stationarity within time proportional to the square of subcube size.
- Infection spreads with positive speed in the studied random environments.
- Infection with recovery also spreads positively and survives indefinitely with positive probability.

## Abstract

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.