Advances in Applied Probability

Local and global survival for nonhomogeneous random walk systems on Z

Daniela Bertacchi, Fábio Prates Machado, and Fabio Zucca

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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).

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Adv. in Appl. Probab., Volume 46, Number 1 (2014), 256-278.

First available in Project Euclid: 1 April 2014

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G50: Sums of independent random variables; random walks

Inhomogeneous random walk frog model egg model local survival global survival


Bertacchi, Daniela; Machado, Fábio Prates; Zucca, Fabio. Local and global survival for nonhomogeneous random walk systems on Z. Adv. in Appl. Probab. 46 (2014), no. 1, 256--278. doi:10.1239/aap/1396360113.

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  • 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.
  • 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.
  • Bertacchi, D. and Zucca, F. (2008). Critical behaviors and critical values of branching random walks on multigraphs. J. Appl. Prob. 45, 481–497.
  • Bertacchi, D. and Zucca, F. (2009). Approximating critical parameters of branching random walks. J. Appl. Prob. 46, 463–478.
  • 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.
  • 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.
  • 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.
  • Gantert, N. and Schmidt, P. (2009). Recurrence for the frog model with drift on $\Z$. Markov Process. Relat. Fields 15 51–58.
  • Junior, V. V., Machado, F. P. and Zuluaga, M. (2011). Rumor processes on $\N$. J. Appl. Prob. 48, 624–636.
  • Lebensztayn, E., Machado, F. P. and Martinez, M. Z. (2010). Nonhomogeneous random walk systems on $\Z$. J Appl. Prob. 47, 562–571.
  • 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.
  • 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.
  • Pemantle, R. (1992). The contact process on trees. Ann. Prob. 20, 2089–2116.
  • 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.
  • Popov, S. Yu. (2001). Frogs in random environment. J. Statist. Phys. 102, 191–201.
  • 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.
  • Telcs, A. and Wormald N. C. (1999). Branching and tree indexed random walks on fractals. J. Appl. Prob. 36, 999–1011.
  • Zucca, F. (2011). Survival, extinction and approximation of discrete-time branching random walks. J. Statist. Phys. 142, 726–753.