Electronic Communications in Probability

The critical density for the frog model is the degree of the tree

Tobias Johnson and Matthew Junge

Full-text: Open access


The frog model on the rooted $d$-ary tree changes from transient to recurrent as the number of frogs per site is increased. We prove that the location of this transition is on the same order as the degree of the tree.

Article information

Electron. Commun. Probab. Volume 21 (2016), paper no. 82, 12 pp.

Received: 28 July 2016
Accepted: 15 November 2016
First available in Project Euclid: 3 December 2016

Permanent link to this document

Digital Object Identifier

Zentralblatt MATH identifier

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)

frog model transience recurrence phase transition trees stochastic dominance

Creative Commons Attribution 4.0 International License.


Johnson, Tobias; Junge, Matthew. The critical density for the frog model is the degree of the tree. Electron. Commun. Probab. 21 (2016), paper no. 82, 12 pp. doi:10.1214/16-ECP29. https://projecteuclid.org/euclid.ecp/1480734227

Export citation


  • [AB05] David J. Aldous and Antar Bandyopadhyay, A survey of max-type recursive distributional equations, Ann. Appl. Probab. 15 (2005), no. 2, 1047–1110.
  • [AMP02] Oswaldo Alves, Fabio Machado, and Serguei Popov, Phase transition for the frog model, Electron. J. Probab. 7 (2002), no. 16, 1–21.
  • [DP14] Christian Döbler and Lorenz Pfeifroth, Recurrence for the frog model with drift on $\mathbb Z^ d$, Electron. Commun. Probab. 19 (2014), no. 79, 13.
  • [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.
  • [HJJ16a] Christopher Hoffman, Tobias Johnson, and Matthew Junge, From transience to recurrence with Poisson tree frogs, Ann. Appl. Probab. 26 (2016), no. 3, 1620–1635.
  • [HJJ16b] Christopher Hoffman, Tobias Johnson, and Matthew Junge, Recurrence and transience for the frog model on trees, to appear in the Annals of Probability, available at arXiv:1404.6238, 2016.
  • [Hoe63] Wassily Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963), 13–30.
  • [JJ16] Tobias Johnson and Matthew Junge, Stochastic orders and the frog model, available at arXiv:1602.04411, 2016.
  • [KZ16] Elena Kosygina and Martin P. W. Zerner, A zero-one law for recurrence and transience of frog processes, to appear in Probability Theory and Related Fields, available at arXiv:1508.01953, 2016.
  • [LP16] Russell Lyons and Yuval Peres, Probability on trees and networks, Cambridge University Press, 2016, Available at http://pages.iu.edu/~rdlyons/.
  • [MSH03] Neeraj Misra, Harshinder Singh, and E. James Harner, Stochastic comparisons of Poisson and binomial random variables with their mixtures, Statist. Probab. Lett. 65 (2003), no. 4, 279–290.
  • [Pop01] Serguei Yu. Popov, Frogs in random environment, J. Statist. Phys. 102 (2001), no. 1-2, 191–201.
  • [Ros16] Josh Rosenberg, The frog model with drift on $\mathbb{R} $, available at arXiv:1605.08414, 2016.
  • [SS07] Moshe Shaked and J. George Shanthikumar, Stochastic orders, Springer Series in Statistics, Springer, New York, 2007.
  • [TW99] András Telcs and Nicholas C. Wormald, Branching and tree indexed random walks on fractals, J. Appl. Probab. 36 (1999), no. 4, 999–1011.
  • [Yu09] Yaming Yu, Stochastic ordering of exponential family distributions and their mixtures, J. Appl. Probab. 46 (2009), no. 1, 244–254.