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Coexistence of lazy frogs on ${\mathbb{Z}}$

Part of: Markov processes Special processes

Published online by Cambridge University Press:  28 June 2022

Mark Holmes Open the ORCID record for Mark Holmes [Opens in a new window]  and
Daniel Kious
Mark Holmes*
Affiliation:
University of Melbourne
Daniel Kious*
Affiliation:
University of Bath
*
*Postal address: School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia. Email: [email protected]
**Postal address: Department of Mathematical Sciences, The University of Bath, Claverton Down, Bath, BA2 7AY. Email: [email protected]
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Abstract

We study the so-called frog model on ${\mathbb{Z}}$ with two types of lazy frogs, with parameters $p_1,p_2\in (0,1]$ respectively, and a finite expected number of dormant frogs per site. We show that for any such $p_1$ and $p_2$ there is positive probability that the two types coexist (i.e. that both types activate infinitely many frogs). This answers a question of Deijfen, Hirscher, and Lopes in dimension one.

Keywords

Frog modelcoexistencerandom walkcompeting growth

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60K37: Processes in random environments
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 3 , September 2022 , pp. 702 - 713
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

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