The Annals of Applied Probability

The shape theorem for the frog model

O. S. M. Alves, F. P. Machado, and S. Yu. Popov

Full-text: Open access

Abstract

We prove a shape theorem for a growing set of simple random walks on $\mathbb{Z}^d$, known as the 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, it becomes active too. At time $0$ all particles are sleeping, except for that placed at the origin. We prove that the set of the original positions of all active particles, rescaled by the elapsed time, converges to some compact convex set.

Article information

Source
Ann. Appl. Probab. Volume 12, Number 2 (2002), 533-546.

Dates
First available in Project Euclid: 17 July 2002

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1026915614

Digital Object Identifier
doi:10.1214/aoap/1026915614

Mathematical Reviews number (MathSciNet)
MR1910638

Zentralblatt MATH identifier
1013.60081

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
Simple random walk subadditive ergodic theorem

Citation

Alves, O. S. M.; Machado, F. P.; Popov, S. Yu. The shape theorem for the frog model. Ann. Appl. Probab. 12 (2002), no. 2, 533--546. doi:10.1214/aoap/1026915614. http://projecteuclid.org/euclid.aoap/1026915614.


Export citation

References