Sensitivity of the frog model to initial conditions

TL;DR

This paper investigates how the recurrence or transience of the frog model on a graph depends on the entire initial distribution of sleeping particles, revealing sensitivity beyond just the mean.

Contribution

It demonstrates that the frog model's behavior is influenced by the full initial distribution, contrasting with related models where only the expectation matters.

Findings

Recurrence or transience depends on the entire initial distribution

Sensitivity contrasts with branching random walk and activated random walk

Distributional properties critically affect the model's dynamics

Abstract

The frog model is an interacting particle system on a graph. Active particles perform independent simple random walks, while sleeping particles remain inert until visited by an active particle. Some number of sleeping particles are placed at each site sampled independently from a certain distribution, and then one particle is activated to begin the process. We show that the recurrence or transience of the model is sensitive not just to the expectation but to the entire distribution. This is in contrast to closely related models like branching random walk and activated random walk.