Existence and sharpness of the phase transition for the frog model on transitive graphs

TL;DR

This paper investigates a modified frog model on various graphs, establishing the existence of phase transitions with respect to key parameters and proving the sharpness of these transitions on transitive graphs.

Contribution

It proves the existence of phase transitions for the model on specific classes of graphs and demonstrates the sharpness of these transitions on transitive graphs.

Findings

Phase transition exists with respect to parameters on non-amenable and quasi-transitive graphs.

Sharpness of phase transition is proven for transitive graphs.

Results apply to graphs with bounded degrees and superlinear polynomial growth.

Abstract

We consider a slight modification of the frog model. For a given graph, each vertex has $\mathrm{Poisson}(\lambda)$ particles (or frogs). At time zero, only the particles at the origin are active, and all the other particles are sleeping. Each active particle performs an independent, continuous-time simple random walk, becoming inactive after time $t$. Once an active frog jumps to a vertex, it activates all of its particles. The survival of active particles can be studied as a dependent percolation model with two parameters $\lambda$ and $t$. In the present work, we establish the existence of a phase transition with respect to each parameter for non-amenable graphs of bounded degrees and quasi-transitive graphs of superlinear polynomial growth, as well as prove the sharpness of the phase transition for transitive graphs.