The shape theorem for the frog model with random initial configuration

2001/10/25 by O. S. M. Alves, Alves, O. S. M., F. P. Machado +7 · 2 citations
Mathematics · #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics #math.PR #msc:60J10 #msc:60K35

paper · pdf · doi:10.48550/arxiv.math/0110280

arxiv created 2001/10/25 · arxiv updated 2009/11/30

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.

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