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

TL;DR

This paper establishes a zero-one law for recurrence and transience in frog models, providing conditions under which these processes almost surely exhibit one of the two behaviors, with applications to various random environments.

Contribution

It introduces sufficient conditions for a zero-one law in frog models, extending to models with i.i.d. frogs and random walks in complex environments.

Findings

Zero-one law for recurrence and transience established.

Conditions for recurrence in elliptic frog processes on ^d.

Applicability to models on super-critical percolation clusters.

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.