Frogs in Random Environment

We study the so-called frog model: Initially there are some sleeping particles and one active particle. A sleeping particle is activated when an active particle hits it, after that the activated particle starts to walk independently of everything and can activate other sleeping particles as well. The initial configuration of sleeping particles is random with density p(x). We identify the critical rate of decay of p(x) separating transience from recurrence, and study some other properties of the model.