Electronic Journal of Probability

The frog model on non-amenable trees

Marcus Michelen and Josh Rosenberg

Full-text: Open access


We examine an interacting particle system on trees commonly referred to as the frog model. For its initial state, it begins with a single active particle at the root and i.i.d. $\mathrm{Poiss} (\lambda )$ many inactive particles at each non-root vertex. Active particles perform discrete time simple random walk and in the process activate any inactive particles they encounter. We show that for every non-amenable tree with bounded degree there exists a phase transition from transience to recurrence (with a non-trivial intermediate phase sometimes sandwiched in between) as $\lambda $ varies.

Article information

Electron. J. Probab., Volume 25 (2020), paper no. 49, 16 pp.

Received: 8 November 2019
Accepted: 7 April 2020
First available in Project Euclid: 28 April 2020

Permanent link to this document

Digital Object Identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

frog model non-amenable interacting random walk

Creative Commons Attribution 4.0 International License.


Michelen, Marcus; Rosenberg, Josh. The frog model on non-amenable trees. Electron. J. Probab. 25 (2020), paper no. 49, 16 pp. doi:10.1214/20-EJP454. https://projecteuclid.org/euclid.ejp/1588039468

Export citation


  • [1] O. S. M. Alves, F. P. Machado, and S. Y. Popov. The shape theorem for the frog model. Ann. Appl. Probab., 12(2):533–546, 2002.
  • [2] C. Döbler and L. Pfeifroth. Recurrence for the frog model with drift on $\mathbb{Z} ^{d}$. Electron. Commun. Probab., 19:no. 79, 13, 2014.
  • [3] N. Gantert and S. Müller. The critical branching Markov chain is transient. Markov Process. Related Fields, 12(4):805–814, 2006.
  • [4] N. Gantert and P. Schmidt. Recurrence for the frog model with drift on $\mathbb{Z} $. Markov Process. Related Fields, 15(1):51–58, 2009.
  • [5] A. Ghosh, S. Noren, and A. Roitershtein. On the range of the transient frog model on $\mathbb{Z} $. Adv. in Appl. Probab., 49(2):327–343, 2017.
  • [6] C. Hoffman, T. Johnson, and M. Junge. From transience to recurrence with Poisson tree frogs. Ann. Appl. Probab., 26(3):1620–1635, 2016.
  • [7] C. Hoffman, T. Johnson, and M. Junge. Recurrence and transience for the frog model on trees. Ann. Probab., 45(5):2826–2854, 2017.
  • [8] T. M. Liggett. Interacting particle systems, volume 276. Springer Science & Business Media, 2012.
  • [9] R. Lyons and Y. Peres. Probability on trees and networks, volume 42 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, New York, 2016.
  • [10] M. Michelen and J. Rosenberg. The frog model on Galton-Watson trees. arXiv preprint arXiv:1910.02367, 2019.
  • [11] S. Y. Popov. Frogs in random environment. J. Statist. Phys., 102(1-2):191–201, 2001.
  • [12] A. F. Ramírez and V. Sidoravicius. Asymptotic behavior of a stochastic combustion growth process. Journal of the European Mathematical Society, 6(3):293–334, 2004.
  • [13] A. Telcs and N. C. Wormald. Branching and tree indexed random walks on fractals. J. Appl. Probab., 36(4):999–1011, 1999.