Electronic Journal of Probability

Recurrence and transience of frogs with drift on $\mathbb{Z} ^d$

Christian Döbler, Nina Gantert, Thomas Höfelsauer, Serguei Popov, and Felizitas Weidner

Full-text: Open access

Abstract

We study the frog model on $\mathbb{Z} ^d$ with drift in dimension $d \geq 2$ and establish the existence of transient and recurrent regimes depending on the transition probabilities. We focus on a model in which the particles perform nearest neighbour random walks which are balanced in all but one direction. This gives a model with two parameters. We present conditions on the parameters for recurrence and transience, revealing interesting differences between dimension $d=2$ and dimension $d \geq 3$. Our proofs make use of (refined) couplings with branching random walks for the transience, and with percolation for the recurrence.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 88, 23 pp.

Dates
Received: 1 September 2017
Accepted: 22 August 2018
First available in Project Euclid: 12 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1536717747

Digital Object Identifier
doi:10.1214/18-EJP216

Mathematical Reviews number (MathSciNet)
MR3858916

Zentralblatt MATH identifier
06964782

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
frog model interacting random walks recurrence transience branching random walk percolation

Rights
Creative Commons Attribution 4.0 International License.

Citation

Döbler, Christian; Gantert, Nina; Höfelsauer, Thomas; Popov, Serguei; Weidner, Felizitas. Recurrence and transience of frogs with drift on $\mathbb{Z} ^d$. Electron. J. Probab. 23 (2018), paper no. 88, 23 pp. doi:10.1214/18-EJP216. https://projecteuclid.org/euclid.ejp/1536717747


Export citation

References

  • [1] O. S. M. Alves, F. P. Machado, and S. Yu. Popov, Phase transition for the frog model, Electron. J. Probab. 7 (2002), no. 16, 21.
  • [2] O. S. M. Alves, F. P. Machado, and S. Yu. Popov, The shape theorem for the frog model, Ann. Appl. Probab. 12 (2002), no. 2, 533–546.
  • [3] B. Bollobás and O. Riordan, Percolation, Cambridge University Press, New York, 2006.
  • [4] C. Döbler and L. Pfeifroth, Recurrence for the frog model with drift on $\mathbb{Z} ^d$, Electron. Commun. Probab. 19 (2014), no. 79.
  • [5] L. R. Fontes, F. P. Machado, and A. Sarkar, The critical probability for the frog model is not a monotonic function of the graph, J. Appl. Probab. 41 (2004), no. 1, 292–298.
  • [6] N. Gantert and S. Müller, The critical branching Markov chain is transient, Markov Process. Related Fields 12 (2006), no. 4, 805–814.
  • [7] N. Gantert and P. Schmidt, Recurrence for the frog model with drift on $\mathbb{Z} $, Markov Process. Related Fields 15 (2009), no. 1, 51–58.
  • [8] A. Ghosh, S. Noren, and A. Roitershtein, On the range of the transient frog model on $\mathbb{Z} $, Adv. in Appl. Probab. 49 (2017), no. 2, 327–343.
  • [9] T. Höfelsauer and F. Weidner, The speed of frogs with drift on $\mathbb Z$, Markov Process. Related Fields 22 (2016), no. 2, 379–392.
  • [10] C. Hoffman, T. Johnson, and M. Junge, Recurrence and transience for the frog model on trees, (2014).
  • [11] C. Hoffman, T. Johnson, and M. Junge, From transience to recurrence with Poisson tree frogs, Ann. Appl. Probab. 26 (2016), no. 3, 1620–1635.
  • [12] T. Johnson and M. Junge, The critical density for the frog model is the degree of the tree, Electron. Commun. Probab. 21 (2016), 12 pp.
  • [13] T. Johnson and M. Junge, Stochastic orders and the frog model, (2016).
  • [14] E. Kosygina and M. P. W. Zerner, A zero-one law for recurrence and transience of frog processes, Probab. Theory Related Fields 168 (2017), no. 1-2, 317–346.
  • [15] G. Lawler and V. Limic, Random walk: a modern introduction, Cambridge Studies in Advanced Mathematics, vol. 123, Cambridge University Press, Cambridge, 2010.
  • [16] É. Lebensztayn, F. P. Machado, and S. Yu. Popov, An improved upper bound for the critical probability of the frog model on homogeneous trees, J. Stat. Phys. 119 (2005), no. 1-2, 331–345.
  • [17] S. Yu. Popov, Frogs in random environment, J. Statist. Phys. 102 (2001), no. 1-2, 191–201.
  • [18] J. Rosenberg, The nonhomogeneous frog model on $\mathbb{Z} $, (2017).
  • [19] A. Telcs and N.C. Wormald, Branching and tree indexed random walks on fractals, J. Appl. Probab. 36 (1999), no. 4, 999–1011.