Sharpness and existence of the Phase Transition for the Frog Model

We consider a small modification of the frog model. For a given vertex-transitive graph, each vertex has $Poisson(\lambda)$ 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. Unlike previous works, we study the survival of active particles as a dependent percolation model with two parameters $\lambda$ and $t$, establishing the existence of the phase transition for certain classes of graphs between the process dying almost surely in finite time and surviving with positive probability. In this talk, we prove the existence of the phase transition for graphs of polynomial growth as well as the sharpness of the phase transition with respect to each of the parameters for transitive graphs. Based on a joint work with Omer Angel, Jonathan Hermon, and Yuliang Shi.