## Electronic Communications in Probability

### Sensitivity of the frog model to initial conditions

#### Abstract

The frog model is an interacting particle system on a graph. Active particles perform independent simple random walks, while sleeping particles remain inert until visited by an active particle. Some number of sleeping particles are placed at each site sampled independently from a certain distribution, and then one particle is activated to begin the process. We show that the recurrence or transience of the model is sensitive not just to the expectation but to the entire distribution. This is in contrast to closely related models like branching random walk and activated random walk.

#### Article information

Source
Electron. Commun. Probab., Volume 24 (2019), paper no. 29, 9 pp.

Dates
Accepted: 12 April 2019
First available in Project Euclid: 5 June 2019

https://projecteuclid.org/euclid.ecp/1559700465

Digital Object Identifier
doi:10.1214/19-ECP230

Mathematical Reviews number (MathSciNet)
MR3962479

Zentralblatt MATH identifier
07068653

#### Citation

Johnson, Tobias; Rolla, Leonardo T. Sensitivity of the frog model to initial conditions. Electron. Commun. Probab. 24 (2019), paper no. 29, 9 pp. doi:10.1214/19-ECP230. https://projecteuclid.org/euclid.ecp/1559700465

#### References

• [1] O. S. M. Alves, F. P. Machado, S. Yu. Popov, and K. Ravishankar, The shape theorem for the frog model with random initial configuration, Markov Process. Related Fields 7 (2001), no. 4, 525–539.
• [2] Riddhipratim Basu, Shirshendu Ganguly, and Christopher Hoffman, Non-fixation for conservative stochastic dynamics on the line, Comm. Math. Phys. 358 (2018), no. 3, 1151–1185.
• [3] J. D. Biggins, Martingale convergence in the branching random walk, J. Appl. Probability 14 (1977), no. 1, 25–37.
• [4] J. D. Biggins, Lindley-type equations in the branching random walk, Stochastic Process. Appl. 75 (1998), no. 1, 105–133.
• [5] Christopher Hoffman, Tobias Johnson, and Matthew Junge, From transience to recurrence with Poisson tree frogs, Ann. Appl. Probab. 26 (2016), no. 3, 1620–1635.
• [6] Christopher Hoffman, Tobias Johnson, and Matthew Junge, Recurrence and transience for the frog model on trees, Ann. Probab. 45 (2017), no. 5, 2826–2854.
• [7] 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.
• [8] Tobias Johnson and Matthew Junge, Stochastic orders and the frog model, Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018), no. 2, 1013–1030.
• [9] Debabrata Panja, Effects of fluctuations on propagating fronts, Physics Reports 393 (2004), no. 2, 87–174.
• [10] Leonardo T. Rolla and Vladas Sidoravicius, Absorbing-state phase transition for driven-dissipative stochastic dynamics on $\mathbb Z$, Invent. Math. 188 (2012), no. 1, 127–150.
• [11] Leonardo T. Rolla, Vladas Sidoravicius, and Olivier Zindy, Universality and sharpness in absorbing-state phase transitions, Ann. Henri Poincaé 20 (2019), no. 6, 1823–1835.
• [12] Vladas Sidoravicius and Augusto Teixeira, Absorbing-state transition for stochastic sandpiles and activated random walks, Electron. J. Probab. 22 (2017), Paper No. 33, 35.
• [13] Alexandre Stauffer and Lorenzo Taggi, Critical density of activated random walks on transitive graphs, Ann. Probab. 46 (2018), no. 4, 2190–2220.