A look at a contact process in random environment
The contact process on a graph S is a toy model for the spread of a population. Each site of S is either occupied or empty: an empty site goes occupied at some growth rate times the number of occupied neighbouring sites while an occupied site independently goes vacant after a unit exponential time. Now, let's add some dynamic (i.e. time-evolving) random environment so that the growth is randomly chosen. We investigate the behaviour of the population with respect to this new environment: one actually exhibits a phase transition phenomenon. After that, we derive the (macroscopic) scaling limit of this system, showing up the mean field equations, corresponding to this underlying (microscopic) random system.