Electronic Communications in Probability

The frog model on trees with drift

Erin Beckman, Natalie Frank, Yufeng Jiang, Matthew Junge, and Si Tang

Full-text: Open access

Abstract

We provide a uniform upper bound on the minimal drift so that the one-per-site frog model on a $d$-ary tree is recurrent. To do this, we introduce a subprocess that couples across trees with different degrees. Finding couplings for frog models on nested sequences of graphs is known to be difficult. The upper bound comes from combining the coupling with a new, simpler proof that the frog model on a binary tree is recurrent when the drift is sufficiently strong. Additionally, we describe a coupling between frog models on trees for which the degree of the smaller tree divides that of the larger one. This implies that the critical drift has a limit as $d$ tends to infinity along certain subsequences.

Article information

Source
Electron. Commun. Probab., Volume 24 (2019), paper no. 26, 10 pp.

Dates
Received: 16 August 2018
Accepted: 18 April 2019
First available in Project Euclid: 1 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1559354659

Digital Object Identifier
doi:10.1214/19-ECP235

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

Keywords
frog model interacting particle system coupling recurrence

Rights
Creative Commons Attribution 4.0 International License.

Citation

Beckman, Erin; Frank, Natalie; Jiang, Yufeng; Junge, Matthew; Tang, Si. The frog model on trees with drift. Electron. Commun. Probab. 24 (2019), paper no. 26, 10 pp. doi:10.1214/19-ECP235. https://projecteuclid.org/euclid.ecp/1559354659


Export citation

References

  • [DGJ$^{+}$17] Micheal Damron, Janko Gravner, Matthew Junge, Hanbaek Lyu, and David Sivakoff, Parking on transitive unimodular graphs, arXiv preprint arXiv:1710.10529 (2017).
  • [DGH$^{+}$17] Christian Döbler, Nina Gantert, Thomas Höfelsauer, Serguei Popov, and Felizitas Weidner, Recurrence and Transience of Frogs with Drift on $\mathbb{Z} ^{d}$, available at arXiv:1709.00038, 2017.
  • [DP14] Christian Döbler and Lorenz Pfeifroth, Recurrence for the frog model with drift on $\mathbb{Z} ^{d}$, Electronic Communications in Probability 19 (2014), 1–13.
  • [FMS04] L. R. Fontes, F. P. Machado, and A. Sarkar, The critical probability for the frog model is not a monotonic function of the graph, Journal of Applied Probability 41 (2004), no. 1, 292–298.
  • [GS09] Nina Gantert and Philipp Schmidt, Recurrence for the frog model with drift on $\mathbb{Z} $, Markov Process. Related Fields 15 (2009), no. 1, 51–58.
  • [GNR17] Arka Ghosh, Steven Noren, and Alexander Roitershtein, On the range of the transient frog model on $\mathbb{Z} $, Adv. in Appl. Probab. 49 (2017), no. 2, 327–343.
  • [HJJ16] Christopher Hoffman, Tobias Johnson, and Matthew Junge, From transience to recurrence with Poisson tree frogs, Ann. Appl. Probab. 26 (2016), no. 3, 1620–1635.
  • [HJJ17a] Christopher Hoffman, Tobias Johnson, and Matthew Junge, Infection spread for the frog model on trees, available at arXiv:1710.05884, 2017.
  • [HJJ17b] Christopher Hoffman, Tobias Johnson, and Matthew Junge, Recurrence and transience for the frog model on trees, Ann. Probab. 45 (2017), no. 5, 2826–2854.
  • [HJJ18] Christopher Hoffman, Tobias Johnson, and Matthew Junge, Cover time for the frog model on trees, available at arXiv:1802.03428, 2018.
  • [JJ16a] Tobias Johnson and Matthew Junge, The critical density for the frog model is the degree of the tree, Electron. Commun. Probab. 21 (2016), Paper No. 82, 12.
  • [JJ16b] Tobias Johnson and Matthew Junge, Stochastic orders and the frog model, to appear in Annales de l’Institut Henri Poincaré, available at arXiv:1602.04411, 2016.
  • [KZ17] Elena Kosygina and Martin P. W. Zerner, A zero-one law for recurrence and transience of frog processes, Probability Theory and Related Fields 168 (2017), no. 1, 317–346.
  • [LMP05] Élcio Lebensztayn, Fábio P Machado, and Serguei Popov, An improved upper bound for the critical probability of the frog model on homogeneous trees, Journal of statistical physics 119 (2005), no. 1-2, 331–345.
  • [RS04] Alejandro F. Ramírez and Vladas Sidoravicius, Asymptotic behavior of a stochastic combustion growth process, J. Eur. Math. Soc. (JEMS) 6 (2004), no. 3, 293–334.
  • [Ros17a] Josh Rosenberg, The frog model with drift on $\mathbb{R} $, Electron. Commun. Probab. 22 (2017), Paper No. 30, 14.
  • [Ros17b] Josh Rosenberg, The nonhomogeneous frog model on $\mathbb{Z} $, Journal of Applied Probability 55 (2017).
  • [Ros17c] Josh Rosenberg, Recurrence of the frog model on the 3,2-alternating tree, Latin American Journal of Probability and Mathematical Statistics 15 (2017).