Sensitivity of the frog model to initial conditions

Tobias Johnson,Leonardo T Rolla

doi:10.1214/19-ecp230

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.

Highlights

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 one particle is activated to begin the process

Summary

Introduction

The frog model is an interacting particle system on a graph. Initially, one designated site contains an active particle, and all other sites contain some number of sleeping particles, typically sampled independently from a given distribution. For two processes resembling the frog model, the long-time behavior of the model depends only on the mean particle distribution The first of these is branching random walk, essentially the frog model except that particles spawn new particles even when moving to a previously visited site. It is a classical result of Biggins’s that for BRW, recurrence vs transience depends only on the mean number of particles spawned. By Theorem 1.1 the same model starting with all but one particle sleeping can fixate with arbitrarily high densities This suggests that the question of fixation vs activity for the ARW is more delicate than it may seem, at least outside the setup of graphs with polynomial growth

Proofs

Generalizations of the argument