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The nonhomogeneous frog model on ℤ

Part of: Markov processes Special processes

Published online by Cambridge University Press:  16 January 2019

Josh Rosenberg
Josh Rosenberg*
Affiliation:
University of Pennsylvania
*
* Current address: School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel. Email address: rjos@math.upenn.edu
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Abstract

We examine a system of interacting random walks with leftward drift on ℤ, which begins with a single active particle at the origin and some distribution of inactive particles on the positive integers. Inactive particles become activated when landed on by other particles, and all particles beginning at the same point possess equal leftward drift. Once activated, the trajectories of distinct particles are independent. This system belongs to a broader class of problems involving interacting random walks on rooted graphs, referred to collectively as the frog model. Additional conditions that we impose on our model include that the number of frogs (i.e. particles) at positive integer points is a sequence of independent random variables which is increasing in terms of the standard stochastic order, and that the sequence of leftward drifts associated with frogs originating at these points is decreasing. Our results include sharp conditions with respect to the sequence of random variables and the sequence of drifts that determine whether the model is transient (meaning the probability infinitely many frogs return to the origin is 0) or nontransient. We consider several, more specific, versions of the model described, and a cleaner, more simplified set of sharp conditions will be established for each case.

Keywords

Frog modeltransiencedriftcoupling

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 4 , December 2018 , pp. 1093 - 1112
Copyright
Copyright © Applied Probability Trust 2018 

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References

[1]Bertacchi, D., Machado, F. P. and Zucca, F. (2012). Local and global survival for nonhomogeneous random walk systems on ℤ. Adv. Appl. Prob. 46, 256–278.Google Scholar
[2]Dobler, C. and Pfeifroth, L. (2014). Recurrence for the frog model with drift on ℤd. Electron. Commun. Prob. 19, 1–13.Google Scholar
[3]Gantert, N. and Schmidt, P. (2009). Recurrence for the frog model with drift on ℤ. Markov Process. Relat. Fields 15, 51–58.Google Scholar
[4]Hoffman, C., Johnson, T. and Junge, M. (2017). Recurrence and transience for the frog model on trees. Ann. Prob. 45, 2826–2854.Google Scholar
[5]Rosenberg, J. (2017). The frog model with drift on ℝ. Electron. Commun. Prob. 22, 1–14.Google Scholar
[6]Telcs, A. and Wormald, N. C. (1989). Branching and tree indexed random walks on fractals. J. Appl. Prob. 36, 999–1011.Google Scholar