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May 2002 The shape theorem for the frog model
O. S. M. Alves, F. P. Machado, S. Yu. Popov
Ann. Appl. Probab. 12(2): 533-546 (May 2002). DOI: 10.1214/aoap/1026915614


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.


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O. S. M. Alves. F. P. Machado. S. Yu. Popov. "The shape theorem for the frog model." Ann. Appl. Probab. 12 (2) 533 - 546, May 2002.


Published: May 2002
First available in Project Euclid: 17 July 2002

zbMATH: 1013.60081
MathSciNet: MR1910638
Digital Object Identifier: 10.1214/aoap/1026915614

Primary: 60K35
Secondary: 60J10

Keywords: Simple random walk , subadditive ergodic theorem

Rights: Copyright © 2002 Institute of Mathematical Statistics


Vol.12 • No. 2 • May 2002
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