The Shape Theorem for the Frog Model with Random Initial Configuration

#### O.S.M. Alves, F.P. Machado, S.Yu. Popov, K. Ravishankar

2001, v.7, №4, 525-539

ABSTRACT

We prove a shape theorem for a growing set of simple random walks (SRWs) 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.

Keywords: subadditive ergodic theorem,simple random walk