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Local and Global Survival for Nonhomogeneous Random Walk Systems on Z

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  22 February 2016

Daniela Bertacchi ,
Fábio Prates Machado  and
Fabio Zucca
Daniela Bertacchi*
Affiliation:
Università di Milano-Bicocca
Fábio Prates Machado*
Affiliation:
Universitade de São Paulo
Fabio Zucca*
Affiliation:
Politecnico di Milano
*
∗ Postal address: Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via Cozzi 53, 20125 Milano, Italy. Email address: daniela.bertacchi@unimib.it
∗∗ Postal address: Instituto de Matemática e Estatística, Universitade de São Paulo, Rua do Matão 1010, São Paulo, Brasil. Email address: fmachado@ime.usp.br
∗∗∗ Postal address: Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy. Email address: fabio.zucca@polimi.it
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Abstract

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We study an interacting random walk system on ℤ where at time 0 there is an active particle at 0 and one inactive particle on each site n ≥ 1. Particles become active when hit by another active particle. Once activated, the particle starting at n performs an asymmetric, translation invariant, nearest neighbor random walk with left-jump probability ln. We give conditions for global survival, local survival, and infinite activation both in the case where all particles are immortal and in the case where particles have geometrically distributed lifespan (with parameter depending on the starting location of the particle). More precisely, once activated, the particle at n survives at each step with probability pn ∈ [0, 1]. In particular, in the immortal case, we prove a 0-1 law for the probability of local survival when all particles drift to the right. Besides that, we give sufficient conditions for local survival or local extinction when all particles drift to the left. In the mortal case, we provide sufficient conditions for global survival, local survival, and local extinction (which apply to the immortal case with mixed drifts as well). Analysis of explicit examples is provided: we describe completely the phase diagram in the cases ½ - ln ~ ± 1 / nα, pn = 1 and ½ - ln ~ ± 1 / nα, 1 - pn ~ 1 / nβ (where α, β > 0).

Keywords

Inhomogeneous random walkfrog modelegg modellocal survivalglobal survival

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60G50: Sums of independent random variables; random walks
Type
Research Article
Information
Advances in Applied Probability , Volume 46 , Issue 1 , March 2014 , pp. 256 - 278
Copyright
© Applied Probability Trust 

References

Alves, O. S. M., Machado, F. P. and Popov, S. Yu. (2002). The shape theorem for the frog model. Ann. Appl. Prob. 12, 533–546.Google Scholar
Alves, O. S. M., Machado, F. P., Popov, S. Yu. and Ravishankar, K. (2001). The shape theorem for the frog model with random initial configuration. Markov Process. Relat. Fields 7, 525–539.Google Scholar
Bertacchi, D. and Zucca, F. (2008). Critical behaviors and critical values of branching random walks on multigraphs. J. Appl. Prob. 45, 481–497.CrossRefGoogle Scholar
Bertacchi, D. and Zucca, F. (2009). Approximating critical parameters of branching random walks. J. Appl. Prob. 46, 463–478.CrossRefGoogle Scholar
Bertacchi, D. and Zucca, F. (2009). Characterization of critical values of branching random walks on weighted graphs through infinite-type branching processes. J. Statist. Phys. 134, 53–65.Google Scholar
Bertacchi, D. and Zucca, F. (2012). Recent results on branching random walks. In Statistical Mechanics and Random Walks: Principles, Processes and Applications, Nova Science Publishers, Hauppauge, NY, pp. 289–340.Google Scholar
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
Gantert, N. and Schmidt, P. (2009). Recurrence for the frog model with drift on Z. Markov Process. Relat. Fields 15 51–58.Google Scholar
Junior, V. V., Machado, F. P. and Zuluaga, M. (2011). Rumor processes on N. J. Appl. Prob. 48, 624–636.Google Scholar
Lebensztayn, E., Machado, F. P. and Martinez, M. Z. (2010). Nonhomogeneous random walk systems on Z. J Appl. Prob. 47, 562–571.Google Scholar
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
Machado, F. P., Menshikov, M. V. and Popov, S. Yu. (2001). Recurrence and transience of multitype branching random walks. Stoch. Process. Appl. 91, 21–37.Google Scholar
Pemantle, R. (1992). The contact process on trees. Ann. Prob. 20, 2089–2116.Google Scholar
Pemantle, R. and Stacey, A. M. (2001). The branching random walk and contact process on Galton–Watson and nonhomogeneous trees. Ann. Prob. 29, 1563–1590.Google Scholar
Popov, S. Yu. (2001). Frogs in random environment. J. Statist. Phys. 102, 191–201.Google Scholar
Popov, S. Yu. (2003). Frogs and some other interacting random walks models. In Discrete Random Walks (Paris, 2003), Association of Discrete Mathematics and Theoretical Computer Science, Nancy, pp. 277–288.Google Scholar
Telcs, A. and Wormald, N.C. (1999). Branching and tree indexed random walks on fractals. J. Appl. Prob. 36, 999–1011.Google Scholar
Zucca, F. (2011). Survival, extinction and approximation of discrete-time branching random walks. J. Statist. Phys. 142, 726–753.Google Scholar