## The Annals of Applied Probability

### From transience to recurrence with Poisson tree frogs

#### Abstract

Consider the following interacting particle system on the $d$-ary tree, known as the frog model: Initially, one particle is awake at the root and i.i.d. Poisson many particles are sleeping at every other vertex. Particles that are awake perform simple random walks, awakening any sleeping particles they encounter. We prove that there is a phase transition between transience and recurrence as the initial density of particles increases, and we give the order of the transition up to a logarithmic factor.

#### Article information

Source
Ann. Appl. Probab., Volume 26, Number 3 (2016), 1620-1635.

Dates
Revised: June 2015
First available in Project Euclid: 14 June 2016

https://projecteuclid.org/euclid.aoap/1465905013

Digital Object Identifier
doi:10.1214/15-AAP1127

Mathematical Reviews number (MathSciNet)
MR3513600

Zentralblatt MATH identifier
1345.60116

#### Citation

Hoffman, Christopher; Johnson, Tobias; Junge, Matthew. From transience to recurrence with Poisson tree frogs. Ann. Appl. Probab. 26 (2016), no. 3, 1620--1635. doi:10.1214/15-AAP1127. https://projecteuclid.org/euclid.aoap/1465905013

#### References

• [1] Aïdékon, E., Hu, Y. and Zindy, O. (2013). The precise tail behavior of the total progeny of a killed branching random walk. Ann. Probab. 41 3786–3878.
• [2] Aldous, D. J. (1991). Random walk covering of some special trees. J. Math. Anal. Appl. 157 271–283.
• [3] Alves, O. S. M., Machado, F. P. and Popov, S. Yu. (2002). The shape theorem for the frog model. Ann. Appl. Probab. 12 533–546.
• [4] Alves, O. S. M., Machado, F. P. and Popov, S. Yu. (2002). Phase transition for the frog model. Electron. J. Probab. 7 no. 16, 21.
• [5] 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. Related Fields 7 525–539.
• [6] Biggins, J. D. (1977). Martingale convergence in the branching random walk. J. Appl. Probab. 14 25–37.
• [7] Comets, F., Quastel, J. and Ramírez, A. F. (2009). Fluctuations of the front in a one dimensional model of $X+Y\to2X$. Trans. Amer. Math. Soc. 361 6165–6189.
• [8] Cranston, M., Mountford, T., Mourrat, J.-C. and Valesin, D. (2014). The contact process on finite homogeneous trees revisited. ALEA Lat. Am. J. Probab. Math. Stat. 11 385–408.
• [9] Dickman, R., Rolla, L. T. and Sidoravicius, V. (2010). Activated random walkers: Facts, conjectures and challenges. J. Stat. Phys. 138 126–142.
• [10] Döbler, C. and Pfeifroth, L. (2014). Recurrence for the frog model with drift on $\mathbb{Z}^{d}$. Electron. Commun. Probab. 19 no. 79, 13.
• [11] Durrett, R. and Liu, X. F. (1988). The contact process on a finite set. Ann. Probab. 16 1158–1173.
• [12] Durrett, R. and Schonmann, R. H. (1988). The contact process on a finite set. II. Ann. Probab. 16 1570–1583.
• [13] Durrett, R., Schonmann, R. H. and Tanaka, N. I. (1989). The contact process on a finite set. III. The critical case. Ann. Probab. 17 1303–1321.
• [14] Gantert, N. and Schmidt, P. (2009). Recurrence for the frog model with drift on $\mathbb{Z}$. Markov Process. Related Fields 15 51–58.
• [15] Ghosh, A. P., Noren, S. and Roitershtein, A. (2015). On the range of the transient frog model on $\mathbb{Z}$. Available at arXiv:1502.02738.
• [16] Hoffman, C., Johnson, T. and Junge, M. (2015). Recurrence and transience for the frog model on trees. Available at arXiv:1404.6238.
• [17] Pemantle, R. (1988). Phase transition in reinforced random walk and RWRE on trees. Ann. Probab. 16 1229–1241.
• [18] Pemantle, R. (1992). The contact process on trees. Ann. Probab. 20 2089–2116.
• [19] Pemantle, R. and Stacey, A. M. (2001). The branching random walk and contact process on Galton–Watson and nonhomogeneous trees. Ann. Probab. 29 1563–1590.
• [20] Popov, S. Yu. (2001). Frogs in random environment. J. Stat. Phys. 102 191–201.
• [21] Popov, S. Yu. (2003). Frogs and some other interacting random walks models. In Discrete Random Walks (Paris, 2003). Discrete Math. Theor. Comput. Sci. Proc., AC 277–288 (electronic). Assoc. Discrete Math. Theor. Comput. Sci., Nancy.
• [22] Ramírez, A. F. and Sidoravicius, V. (2004). Asymptotic behavior of a stochastic combustion growth process. J. Eur. Math. Soc. (JEMS) 6 293–334.
• [23] Rolla, L. T. and Sidoravicius, V. (2012). Absorbing-state phase transition for driven-dissipative stochastic dynamics on ${\mathbb{Z}}$. Invent. Math. 188 127–150.
• [24] Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.
• [25] Sidoravicius, V. and Teixeira, A. (2014). Absorbing-state transition for Stochastic Sandpiles and Activated Random Walks. Available at arXiv:1412.7098.
• [26] Telcs, A. and Wormald, N. C. (1999). Branching and tree indexed random walks on fractals. J. Appl. Probab. 36 999–1011.