The asymptotic shape theorem for the frog model on finitely generated abelian groups

Cristian F. Coletti, Lucas R. de Lima

Published 2019-08-27Version 1

We study the frog model on Cayley graphs of groups with polynomial growth rate $D \geq 3$. We prove that the activation time of particles grow at least linearly and we show the asymptotic shape theorem in the abelian case with any finite generator set. The frog model describes an interacting particle system in discrete time. We consider that the process begins with a particle at each vertex of the graph and only one of these particles is active. The active particle jumps to a neighboring site in an equiprobable way and activates the particle that was already there. Each activated particle performs a simple random walk in discrete time activating the inactive particles in the visited vertices.