A zero-one law for recurrence and transience of frog processes

Abstract

We provide sufficient conditions for the validity of a dichotomy, i.e. zero-one law, between recurrence and transience of general frog models. In particular, the results cover frog models with i.i.d. numbers of frogs per site where the frog dynamics are given by quasi-transitive Markov chains or by random walks in a common random environment including super-critical percolation clusters on $\mathbb{Z}^d$. We also give a sufficient and almost sharp condition for recurrence of uniformly elliptic frog processes on $\mathbb{Z}^d$. Its proof uses the general zero-one law.