Advances in Applied Probability

Local and global survival for nonhomogeneous random walk systems on Z

Daniela Bertacchi, Fábio Prates Machado, and Fabio Zucca

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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).

Article information

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

Dates
First available in Project Euclid: 1 April 2014

Permanent link to this document
https://projecteuclid.org/euclid.aap/1396360113

Digital Object Identifier
doi:10.1239/aap/1396360113

Mathematical Reviews number (MathSciNet)
MR3189058

Zentralblatt MATH identifier
1302.60131

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

Keywords
Inhomogeneous random walk frog model egg model local survival global survival

Citation

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. https://projecteuclid.org/euclid.aap/1396360113


Export citation

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.
  • 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.