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The critical probability for the frog model is not a monotonic function of the graph

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

L. R. Fontes ,
F. P. Machado  and
A. Sarkar
L. R. Fontes*
Affiliation:
University of São Paulo
F. P. Machado*
Affiliation:
University of São Paulo
A. Sarkar*
Affiliation:
Indian Statistical Institute, Delhi
*
∗ Postal address: Department of Statistics, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão 1010, CEP 05508-900, São Paulo SP, Brazil.
∗ Postal address: Department of Statistics, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão 1010, CEP 05508-900, São Paulo SP, Brazil.
∗∗∗ Postal address: Indian Statistical Institute, Delhi Centre, 7 S. J. S. Sansanwal Marg, New Delhi, 110016, India.
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Abstract

We show that the critical probability for the frog model on a graph is not a monotonic function of the graph. This answers a question of Alves, Machado and Popov. The nonmonotonicity is unexpected as the frog model is a percolation model.

Keywords

Frog modelphase transitioncritical probability

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Short Communications
Information
Journal of Applied Probability , Volume 41 , Issue 1 , March 2004 , pp. 292 - 298
Copyright
Copyright © Applied Probability Trust 2004 

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References

Alves, O. S. M., Machado, F. P. and Popov, S. Yu. (2002). Phase transition for the frog model. Electron. J. Prob. 7, No. 16.CrossRefGoogle Scholar
Benjamini, I., and Schramm, O. (1996). Percolation beyond ℤd, many questions and a few answers. Electron. Commun. Prob. 1, 71–82.CrossRefGoogle Scholar
Durrett, R. (1988). Lecture Notes on Particle Systems and Percolation. Wadsworth and Brooks, Pacific Grove, CA.Google Scholar
Grimmett, G. (1991). Strict monotonicity for critical points in percolation and ferromagnetic models. J. Statist. Phys. 63, 817–835.Google Scholar