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

TL;DR

This paper proves a shape theorem for the frog model on finitely generated abelian groups, showing linear growth of activation time and a limiting shape of activated sites on Cayley graphs with polynomial growth.

Contribution

It establishes the asymptotic shape theorem for the frog model on abelian groups, extending understanding of activation dynamics in these systems.

Findings

Activation time grows at least linearly

Set of activated sites converges to a limiting shape

Results apply to Cayley graphs with polynomial growth D ≥ 3

Abstract

We study the frog model on Cayley graphs of groups with polynomial growth rate $D \geq 3$. The frog model is 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 when the process begins. Each activated particle performs a simple random walk in discrete time activating the inactive particles in the visited vertices. We prove that the activation time of particles grows at least linearly and we show that in the abelian case with any finite generator set the set of activated sites has a limiting shape.