Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T14:09:34.937Z Has data issue: false hasContentIssue false
  • English
  • Français

Frog models on trees through renewal theory

Part of: Time-dependent statistical mechanics (dynamic and nonequilibrium) Special processes

Published online by Cambridge University Press:  16 November 2018

Sandro Gallo  and
Pablo M. Rodriguez
Sandro Gallo*
Affiliation:
UFSCar
Pablo M. Rodriguez*
Affiliation:
USP
*
* Postal address: Departamento de Estatística, UFSCar, Rodovia Washington Luiz, km 235, CEP 13565-905, São Carlos, SP, Brasil. Email address: sandrodobrasil@gmail.com
** Postal address: Instituto de Ciências Matemáticas e de Computação, USP, Av. Trabalhador são-carlense 400 - Centro, CEP 13560-970, São Carlos, SP, Brasil.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a class of growing systems of random walks on regular trees, known as frog models with geometric lifetime in the literature. With the help of results from renewal theory, we derive new bounds for their critical parameters. Our approach also improves the existing bounds for the critical parameter of a percolation model on trees known as cone percolation.

Keywords

Frog modelcritical probabilitycone percolationhomogeneous treerenewal theory

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60K05: Renewal theory 82C41: Dynamics of random walks, random surfaces, lattice animals, etc.
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 3 , September 2018 , pp. 887 - 899
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Alves, O. S. M., Machado, F. P. and Popov, S. Y. (2002). The shape theorem for the frog model. Ann. Appl. Prob. 12, 533–546.Google Scholar
[2]Alves, O. S. M., Lebensztayn, E., Machado, F. P. and Martinez, M. Z. (2006). Random walks systems on complete graphs. Bull. Braz. Math. Soc. (N. S.) 37, 571–580.Google Scholar
[3]Brémaud, P. (1999). Markov Chains (Texts Appl. Math. 31). Springer, New York.Google Scholar
[4]Bressaud, X., Fernández, R. and Galves, A. (1999). Decay of correlations for non-Hölderian dynamics. A coupling approach. Electron. J. Prob. 4, 3.Google Scholar
[5]Fontes, L. R., Machado, F. P. and Sarkar, A. (2004). The critical probability for the frog model is not a monotonic function of the graph. J. Appl. Prob. 41, 292–298.Google Scholar
[6]Gallo, S. and Rodríguez, P. M. (2017). Frog models on trees through renewal theory. Preprint. Available at https://arxiv.org/abs/1705.00111v1.Google Scholar
[7]Gallo, S., Garcia, N. L., Junior, V. V. and Rodríguez, P. M. (2014). Rumor processes on ℕ and discrete renewal processes. J. Statist. Phys. 155, 591–602.Google Scholar
[8]Giacomin, G. (2008). Renewal convergence rates and correlation decay for homogeneous pinning models. Electron. J. Prob. 13, 513–529.Google Scholar
[9]Hoffman, C., Johnson, T. and Junge, M. (2017). Recurrence and transience for the frog model on trees. Ann. Prob. 45, 2826–2854.Google Scholar
[10]Junior, V. V., Machado, F. P. and Zuluaga, M. (2011). Rumor processes on ℕ. J. Appl. Prob. 48, 624–636.Google Scholar
[11]Junior, V. V., Machado, F. P. and Zuluaga, M. (2014). The cone percolation on 𝕋d. Braz. J. Prob. Statist. 28, 367–375.Google Scholar
[12]Kurtz, T. G., Lebensztayn, E., Leichsenring, A. R. and Machado, F. P. (2008). Limit theorems for an epidemic model on the complete graph. ALEA Latin Amer. J. Prob. Math. Statist. 4, 45–55.Google Scholar
[13]Lebensztayn, E. and Rodriguez, P. M. (2013). A connection between a system of random walks and rumor transmission. Phys. A 392, 5793–5800.Google Scholar
[14]Lebensztayn, E., Machado, F. P. and Martinez, M. Z. (2006). Self-avoiding random walks on homogeneous trees. Markov Process. Relat. Fields 12, 735–745.Google Scholar
[15]Lebensztayn, E., Machado, F. P. and Martinez, M. Z. (2016). Random walks systems with finite lifetime on ℤ. J. Statist. Phys. 162, 727–738.Google Scholar
[16]Lebensztayn, É., Machado, F. P. and Popov, S. (2005). An improved upper bound for the critical probability of the frog model on homogeneous trees. J. Statist. Phys. 119, 331–345.Google Scholar
[17]Lebensztayn, E., Machado, F. P. and Rodríguez, P. M. (2011). On the behaviour of a rumour process with random stifling. Environ. Model. Software 26, 517–522.Google Scholar
[18]Popov, S. Y. (2003). Frogs and some other interacting random walks models. In Discrete Random Walks, Association of Discrete Mathematics and Theoretical Computer Science, Nancy, pp. 277–288.Google Scholar
[19]Silvestrov, D. and Petersson, M. (2013). Exponential expansions for perturbed discrete time renewal equations. In Applied Reliability Engineering and Risk Analysis: Probabilistic Models and Statistical Inference, John Wiley, Chichester, pp. 349–362.Google Scholar