The nonhomogeneous frog model on $\mathbb{Z}$

TL;DR

This paper analyzes a variant of the frog model on the integer lattice, establishing precise conditions under which the system is transient or recurrent based on particle distribution and drift parameters.

Contribution

It provides sharp, generalized criteria for transience and recurrence in a nonhomogeneous frog model with variable particle counts and drifts on .

Findings

Derived conditions for transience and recurrence based on particle distribution and drift sequences.

Established sharp thresholds for model behavior in various specific cases.

Extended results to simplified versions of the frog model.

Abstract

We examine a system of interacting random walks with leftward drift on $\mathbb{Z}$, which begins with a single active particle at the origin and some distribution of inactive particles on the positive integers. Inactive particles become activated when landed on by other particles, and all particles beginning at the same point posses equal leftward drift. Once activated, the trajectories of distinct particles are independent. This system belongs to a broader class of problems involving interacting random walks on rooted graphs, referred to collectively as the frog model. Additional conditions that we impose on our model include that the number of frogs (i.e. particles) at positive integer points is a sequence of independent random variables which is increasing in terms of the standard stochastic order, and that the sequence of leftward drifts associated with frogs originating at these points is decreasing. Our results include sharp conditions with respect to the sequence of random variables and the sequence of drifts, that determine whether the model is transient (meaning the probability infinitely many frogs return to the origin is $0$) or non-transient. Several, more specific, versions of the model described will also be considered, and a cleaner, more simplified set of sharp conditions will be established for each case.