## Electronic Journal of Probability

### Cluster growth in the dynamical Erdős-Rényi process with forest fires

#### Abstract

We investigate the growth of clusters within the forest fire model of Ráth and Tóth [EJP, vol 14, paper no 45]. The model is a continuous-time Markov process, similar to the dynamical Erdős-Rényi random graph but with the addition of so-called fires. A vertex may catch fire at any moment and, when it does so, causes all edges within its connected cluster to burn, meaning that they instantaneously disappear. Each burned edge may later reappear.

We give a precise description of the process $C_t$ of the size of the cluster of a tagged vertex, in the limit as the number of vertices in the model tends to infinity. We show that $C_t$ is an explosive branching process with a time-inhomogeneous offspring distribution and instantaneous return to 1 on each explosion. Additionally, we show that the characteristic curves used to analyse the Smoluchowski-type coagulation equations associated to the model have a probabilistic interpretation in terms of the process $C_t$.

#### Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 101, 33 pp.

Dates
Accepted: 24 September 2015
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465067207

Digital Object Identifier
doi:10.1214/EJP.v20-4035

Mathematical Reviews number (MathSciNet)
MR3407218

Zentralblatt MATH identifier
1328.60208

Rights

#### Citation

Crane, Edward; Freeman, Nic; Tóth, Bálint. Cluster growth in the dynamical Erdős-Rényi process with forest fires. Electron. J. Probab. 20 (2015), paper no. 101, 33 pp. doi:10.1214/EJP.v20-4035. https://projecteuclid.org/euclid.ejp/1465067207

#### References

• Ahlberg, Daniel; Duminil-Copin, Hugo; Kozma, Gady; Sidoravicius, Vladas. Seven-dimensional forest fires. Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015), no. 3, 862–866.
• Aldous, David J. The percolation process on a tree where infinite clusters are frozen. Math. Proc. Cambridge Philos. Soc. 128 (2000), no. 3, 465–477.
• Aldous, David J. Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5 (1999), no. 1, 3–48.
• van den Berg, J.; Brouwer, R. Self-destructive percolation. Random Structures Algorithms 24 (2004), no. 4, 480–501.
• van den Berg, J.; Brouwer, R. Self-organized forest-fires near the critical time. Comm. Math. Phys. 267 (2006), no. 1, 265–277.
• van den Berg, J.; Tóth, B. A signal-recovery system: asymptotic properties, and construction of an infinite-volume process. Stochastic Process. Appl. 96 (2001), no. 2, 177–190.
• Bertoin, Jean. Burning cars in a parking lot. Comm. Math. Phys. 306 (2011), no. 1, 261–290.
• Bressaud, Xavier; Fournier, Nicolas. Asymptotics of one-dimensional forest fire processes. Ann. Probab. 38 (2010), no. 5, 1783–1816.
• Bressaud, Xavier; Fournier, Nicolas. A mean-field forest-fire model. ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014), no. 1, 589–614.
• Deaconu, Madalina; Fournier, Nicolas; Tanré, Etienne. A pure jump Markov process associated with Smoluchowski's coagulation equation. Ann. Probab. 30 (2002), no. 4, 1763–1796.
• B. Drossel and F. Schwabl. Self-organized critical forest fires model. Physical Review Letters, 69:1629–1632, 1992.
• Dürre, Maximilian. Existence of multi-dimensional infinite volume self-organized critical forest-fire models. Electron. J. Probab. 11 (2006), no. 21, 513–539 (electronic).
• Dürre, Maximilian. Uniqueness of multi-dimensional infinite volume self-organized critical forest-fire models. Electron. Comm. Probab. 11 (2006), 304–315 (electronic).
• M. Dürre. Self-organized Critical Phenomena; Forest Fire and Sandpile Model. PhD thesis, Ludwig-Maximilians-Universität München, 2009.
• Fournier, Nicolas; Laurençot, Philippe. Marcus-Lushnikov processes, Smoluchowski's and Flory's models. Stochastic Process. Appl. 119 (2009), no. 1, 167–189.
• R. Graf. Self-destructive percolation as a limit of forest-fire models on regular rooted trees. arXiv:1404.0325, 2014.
• Graf, Robert. A forest-fire model on the upper half-plane. Electron. J. Probab. 19 (2014), no. 8, 27 pp.
• D. Kiss, I. Manolescu, and V. Sidoravicius. Planar lattices do not recover from forest fires. Annals of Probability (to appear) arXiv:1312.7004, 2013.
• M. Merle and R. Normand. Self-organized criticality in a discrete model for Smoluchowski's equation. arXiv:1410.8338, 2014.
• Norris, James R. Smoluchowski's coagulation equation: uniqueness, nonuniqueness and a hydrodynamic limit for the stochastic coalescent. Ann. Appl. Probab. 9 (1999), no. 1, 78–109.
• Norris, J. R. Cluster coagulation. Comm. Math. Phys. 209 (2000), no. 2, 407–435.
• G. Preussner. Self-Organized Criticality - Theory, Models and Characterisation. Cambridge University Press, 2012.
• Ráth, Balázs. Mean field frozen percolation. J. Stat. Phys. 137 (2009), no. 3, 459–499.
• Ráth, Balézs; Táth, Bálint. Erdós-Rényi random graphs $+$ forest fires $=$ self-organized criticality. Electron. J. Probab. 14 (2009), no. 45, 1290–1327.
• K. Schenk, B. Drossel, and Schwabl. The self-organized critical forest-fire model on large scales. Physical Review E, 65:026135, 2002.
• A. Stahl. Existence of a stationary distribution for multi-dimensional infinite volume forest-fire processes. arXiv:1203.5533v1, 2012.