The shape theorem for the frog model

By O.S.M.Alves, S.Yu.Popov, F.P.Machado. 2001

125 citations recorded by the source corpus.

We prove a shape theorem for a growing set of simple random walks on Zd, 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.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b2ec210d3ed609e264dd59eecf0363048f31a5cc

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In Condensed Matter Physics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

On multidimensional branching random walks in random environment (2005, background): “From this point on, the proof of the shape result for B0 n becomes completely standard, so we only briefly outline the main steps and refer to, for example, [1, 5, 9] for details:”
The spread of a rumor or infection in a moving population (2003, unknown): “ar X iv :m at h/ 03 12 49 6v 2 [ m at h. PR ] 7 F eb 2 00 6 The Annals of Probability 2005, Vol. 33, No. 6, 2402–2462 DOI: 10.1214/009117905000000413 c© Institute of Mathematical Statistics, 2005”
A shape theorem for the spread of an infection (2003, background): “This exception is the so-called frog model which follows the rules given in our abstract, but which has DA = 0, i.e., the susceptibles or type A cells stand still (see [ AMP ] and [RS] for this model).”