Existence and Sharpness of the Phase Transition for the Frog Model on Transitive Graphs

The frog model is an interacting particle system that describes the spread of an infection or rumor through a moving population. In the version we consider, the model depends on two parameters, λ and t. On a given graph, each vertex initially contains a Poisson(λ) number of particles (frogs). At time zero, only the particles at the origin are infected, while all others are susceptible. Infected particles perform independent, continuous-time simple random walks, whereas susceptible particles remain stationary until they become infected. Whenever an infected particle encounters a susceptible one, the susceptible particle becomes infected instantaneously. In addition, each infected particle has a finite lifetime t, after which it is removed from the system.

 

We show that, for several classes of transitive graphs, including graphs of polynomial growth and non-amenable graphs, this model exhibits a phase transition between the extinction and survival of infected particles in both parameters. We also prove that this phase transition is necessarily sharp on every vertex-transitive graph. Joint work with Omer Angel, Daniel de la Riva Massaad, and Jonathan Hermon.