Description:

Title: Sharpness and existence of the Phase Transition for the Frog Model.

Abstract: 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.