The shape theorem for the frog model with random initial configuration
Probability
2007-05-23 v1
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.
Cite
@article{arxiv.math/0110280,
title = {The shape theorem for the frog model with random initial configuration},
author = {O. S. M. Alves and F. P. Machado and S. Yu. Popov and K. Ravishankar},
journal= {arXiv preprint arXiv:math/0110280},
year = {2007}
}