The shape theorem for the frog model with random initial configuration

TL;DR

This paper establishes a shape theorem for the frog model on Z^d, showing that the scaled set of visited sites converges to a convex shape, advancing understanding of growth processes with random initial conditions.

Contribution

It proves a shape theorem for the frog model with random initial configurations, extending previous results to more general initial distributions.

Findings

Visited sites form a convex shape upon scaling

Convergence to a deterministic shape is proven

Results apply to models with random initial particle distributions

Abstract

We prove a shape theorem for a growing set of simple random walks on Z^d, known as frog model. The dynamics of this process is described as follows: There are active particles, which perform independent discrete time SRWs, and sleeping particles, which do not move. When a sleeping particle is hit by an active particle, the former becomes active as well. Initially, a random number of particles is placed into each site. At time 0 all particles are sleeping, except for those placed at the origin. We prove that the set of all sites visited by active particles, rescaled by the elapsed time, converges to a compact convex set.