One-dimensional general forest fire processes
Mémoires de la Société Mathématique de France, no. 132 (2013) , 144 p.
The full text of recent articles is available to journal subscribers only. See the article on the journal's website.

We consider the one-dimensional generalized forest fire process: at each site of , seeds and matches fall according to i.i.d. stationary renewal processes. When a seed falls on an empty site, a tree grows immediately. When a match falls on an occupied site, a fire starts and destroys immediately the corresponding connected component of occupied sites. Under some quite reasonable assumptions on the renewal processes, we show that when matches become less and less frequent, the process converges, with a correct normalization, to a limit forest fire model. According to the nature of the renewal processes governing seeds, there are four possible limit forest fire models. The four limit processes can be perfectly simulated. This study generalizes consequently previous results of [15] where seeds and matches were assumed to fall according to Poisson processes.

Nous étudions le processus des feux de forêt généralisé en dimension 1 : sur chaque site de , des graines et des allumettes tombent suivant des processus de renouvellement stationnaires i.i.d. Quand une graine tombe sur un site vide, un arbre pousse immédiatement. Quand une allumette tombe sur un site occupé, un feu démarre et brûle immédiatement la composante connexe occupée autour de ce site. Nous montrons — sous des hypothèses raisonnables sur les processus de renouvellement — que lorsque la fréquence des allumettes tend vers zéro, le processus converge, correctement renormalisé, vers un processus limite. Suivant la nature des processus de renouvellement gouvernant l’apparition des graines, quatre processus limites sont possibles. Les quatre modèles limites peuvent être simulés parfaitement. Cette étude généralise des résultats de [15], où nous supposions que graines et allumettes tombaient suivant des processus de Poisson.

DOI: 10.24033/msmf.442
Classification: 60K35,  82C22
Keywords: Stochastic interacting particle systems, Self-organized criticality, Forest fire model
     author = {Bressaud, Xavier and Fournier, Nicolas},
     title = {One-dimensional general forest fire processes},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {132},
     year = {2013},
     doi = {10.24033/msmf.442},
     zbl = {1297.60063},
     language = {en},
     url = {}
AU  - Bressaud, Xavier
AU  - Fournier, Nicolas
TI  - One-dimensional general forest fire processes
T3  - Mémoires de la Société Mathématique de France
PY  - 2013
IS  - 132
PB  - Société mathématique de France
UR  -
UR  -
UR  -
DO  - 10.24033/msmf.442
LA  - en
ID  - MSMF_2013_2_132__1_0
ER  - 
%0 Book
%A Bressaud, Xavier
%A Fournier, Nicolas
%T One-dimensional general forest fire processes
%S Mémoires de la Société Mathématique de France
%D 2013
%N 132
%I Société mathématique de France
%R 10.24033/msmf.442
%G en
%F MSMF_2013_2_132__1_0
Bressaud, Xavier; Fournier, Nicolas. One-dimensional general forest fire processes. Mémoires de la Société Mathématique de France, Serie 2, no. 132 (2013), 144 p. doi : 10.24033/msmf.442.

[1] Aldous (D. J.)Emergence of the giant component in special Marcus-Lushnikov processes, Random Structures Algorithms, t.12 (1998), pp.179–196. | Zbl

[2] —, Deterministic and stochastic models for coalescence (aggregation, coagulation): A review of the mean-field theory for probabilists, Bernoulli, t.5 (1999), pp.3–48. | Zbl

[3] —, The percolation process on a tree where infinite clusters are frozen, Math. Proc. Cambridge Philos. Soc., t.128 (2000), pp.465–477. | Zbl

[4] Amaral (L .A. N.) & Lauritsen (K. B.) – Self-organized criticality in a rice-pile model, Phys Rev. E., t.54 (1996), pp.4512–4515.

[5] Bak (P.), Tang (C.) & Wiesenfeld (K.)Self-organized criticality: an explanation of 1/f noise, Phys. Rev. Letters, t.59 (1987), pp.381–384.

[6] —, Self-organized criticality, Phys. Rev. A, t.38 (1988), pp.364–374. | Zbl

[7] Van Den Berg (J.) & Brouwer (R.)Self-organized forest-fires near the critical time, Comm. Math. Phys., t.267 (2006), pp.265–277. | Zbl

[8] Van Den Berg (J.), Brouwer (R.) & Vágvölgyi (B.)Box-crossings and continuity results for self-destructive percolation in the plane, pp. 117–136 in In and Out of Equilibrium 2, Progress in Probability, vol.60, Birkhäuser, 2008. | Zbl

[9] Van Den Berg (J.) & Járai (A. A.)On the asymptotic density in a one-dimensional self-organized critical forest-fire model, Comm. Math. Phys., t.253 (2005), pp.633–644. | Zbl

[10] Van Den Berg (J.) & Tóth (B.)A signal-recovery system: asymptotic properties, and construction of an infinite-volume process, Stochastic Process. Appl., t.96 (2001), pp.177–190. | MR | Zbl

[11] Bertoin (J.)Random fragmentation and coagulation processes, Cambridge Studies in Advanced Math., t.102 (2006). | Zbl

[12] —, Burning cars in parkings, preprint, http://hal.archives-ouvertes. fr/ccsd-00505206/en/, 2010.

[13] Borgne (Y. Le) & Rossin (D.) – On the identity of the sand pile group, Discrete Math., t.256 (2002), pp.775–790.

[14] Bressaud (X.) & Fournier (N.)On the invariant distribution of an avalanche process, Ann. Probab., t.37 (2009), pp.48–77. | Zbl

[15] —, Asymptotics of one-dimensional forest fire processes., Ann. Probab., t.38 (2010), pp.1783–1816. | Zbl

[16] Brouwer (R.)A modified version of frozen percolation on the binary tree, preprint, 2005, arXiv: math/0511021v1.

[17] Brouwer (R.) & Pennanen (J.)The cluster size distribution for a forest-fire process on , Electron. J. Prob., t.11 (2006), pp.1133–1143. | Zbl | EuDML

[18] Brylawski (T.)The lattice of integer partitions, Discrete Math., t.6 (1973), pp.201–219. | Zbl

[19] Cafiero (R.), Pietronero (L.) & Vespignani (A.)Persistence of screening and self-criticality in the scale invariant dynamics of diffusion limited aggregation, Phys. Rev. Lett., t.70 (1993), pp.3939–3942.

[20] Christensen (K.), Corral (A.), Frette (V.), Feder (J.) & Jossang (T.)Tracer dispersion in a self-organized critical system, Phys. Rev. Lett., t.77 (1996), pp.107–110.

[21] Cocozza-Thivent (C.)Processus stochastiques et fiabilité des systèmes, Springer-Smai, 1997.

[22] Corral (A.), Telesca (L.) & Lasaponara (R.)Scaling and correlations in the dynamics of forest-fire occurrence, Phys. Rev. E, t.77 (2008), 016101, 7 pp.

[23] Cui (W.) & Perera (A. H.)What do we know about forest fire size distribution, and why is this knowledge useful for forest management?, Internat. J. Wildland Fire, t.17 (2008), pp.234–244.

[24] Damron (M.), Sapozhnikov (A.) & Vágvölgyi (B.)Relations between invasion percolation and critical percolation in two dimensions, Ann. Probab., t.37 (2009), pp.2297–2331. | Zbl

[25] Dhar (D.)Self-organized critical state of sand pile automaton models, Phys. Rev. Lett., t.64 (1990), pp.1613–1616. | Zbl

[26] —, Theoretical studies of self-organized criticality, Phys. A, t.369 (2006), pp.29–70.

[27] Drossel (B.) & Schwabl (F.)Self-organized critical forest-fire model, Phys. Rev. Lett., t.69 (1992), pp.1629–1632.

[28] Drossela (B.), Clar (S.) & Schwabl (F.)Exact results for the one-dimensional self-organized critical forest-fire model, Phys. Rev. Lett., t.71 (1993), pp.3739–3742.

[29] Dürre (M.)Existence of multi-dimensional volume self-organized critical forest-fire models, Electron. J. Prob., t.11 (2006), pp.513–539. | Zbl | EuDML

[30] —, Uniqueness of multi-dimensional infinite volume self-organized critical forest-fire models, Electronic Comm. Prob., t.11 (2006), pp.304–315. | Zbl | EuDML

[31] —, Self-organized critical phenomena: Forest fire and sand pile models, Thèse, LMU München, 2009.

[32] Erdős (P.) & Rényi (A.)On the evolution of random graphs, Bull. Inst. Internat. Statist., t.38 (1961), pp.343–347. | Zbl

[33] Fournier (N.) & Laurençot (P.)Marcus Lushnikov processes, Smoluchowski’s and Flory’s models, Stochastic Process. Appl., t.119 (2009), pp.167–189. | Zbl

[34] Goles (E.), Latapy (M.), Magnien (C.), Morvan (M.) & Phan (H. D.)Sand pile models and lattices: a comprehensive survey, Theoret. Comput. Sci., t.322 (2004), pp.383–407. | Zbl

[35] Grassberger (P.)Critical behaviour of the Drossel-Schwabl forest fire model, New J. Physics, t.4 (2002), pp.17.

[36] Grimmett (G.)Percolation, Springer-Verlag, New York, 1989. | Zbl

[37] Henley (C. L.)Self-organized percolation: a simpler model, Bull. Am. Phys. Soc., t.34 (1989), pp.838.

[38] Holmes (T. P.), Huggett (R. J.) & Westerling (A. L.)Statistical analysis of large wildfires, pp. 59–77 in The Economics of Forest Disturbances: Wildfires, Storms, and Invasive Species, Springer, Dordrecht, 2008.

[39] Holroyd (A. E.), Levine (L.), Meszaros (K.), Y. Peres (J. Propp) & Wilson (D.) – Chip-Firing and Rotor-Routing on Directed Graphs, pp. 331–364 in In and Out of Equilibrium 2, Progr. Probab., vol.60, 2008.

[40] Honecker (A.) & Peschel (I.)Critical properties of the one-dimensional forest-fire model, Physica A, t.229 (1996), pp.478–500.

[41] Jacod (J.) & Shiryaev (A. N.)Limit theorems for stochastic processes, Springer, 1982.

[42] Járai (A. A.)Thermodynamic limit of the abelian sand pile model on d , Markov Process. Related Fields, t.11 (2005), pp.313–336.

[43] Jensen (H. J.)Self-organized criticality, Cambridge Lecture Notes in Physics, vol.10, Cambridge University Press, 1998.

[44] Kingman (J.)The coalescent, Stochastic Process. Appl., t.13 (1982), pp.235–248. | Zbl

[45] Liggett (T. M.)Interacting particle systems, Springer, 1985. | Zbl

[46] Lübeck (S.) & Usadel (K. D.)The Bak-Tang-Wiesenfeld sand pile model around the upper critical dimension, Phys. Rev. E, t.56 (1997), pp.5138–5143.

[47] Maes (C.), Redig (F.) & Saada (E.)The infinite volume limit of dissipative abelian sand piles, Commun. Math. Phys., t.244 (2004), pp.395–417. | Zbl

[48] Majumdar (S. N.)Exact fractal dimension of the loop-erased random walk in two dimensions, Phys. Rev. Lett., t.68 (1992), pp.2329–2331.

[49] Mangiavillano (A.)Multi-scalarité du phénomène feu de forêt en régions méditerranéennes françaises de 1973 à 2006, Thèse, Université d’Avignon, 2008.

[50] Olami (Z.), Feder (H. J. S.) & Christensen (K.)Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes, Physical Review Letters, t.68 (1992), pp.1244–1247.

[51] Pietronero (L.), Tartaglia (P.) & Zhang (Y. C.)Theoritical studies of self-organized criticality, Physica A, t.173 (1991), pp.22–44.

[52] Priezzhev (V. B.), Dhar (D.), Dhar (A.) & Krishnamurthy (S.)Eulerian walkers as a model of self-organised criticality, Phys. Rev. Lett., t.77 (1996), pp.5079–5082.

[53] Pruessner (G.) & Jensen (H. J.)A new, efficient algorithm for the forest fire model, 2003.

[54] Ráth (B.)Mean field frozen percolation, J. Statist. Physics, t.137 (2009), pp.459–499. | Zbl

[55] Ráth (B.) & Tóth (B.)Erdős-Rényi random graphs + forest fires = self-organized criticality, Electronic J. Probab., t.14 (2009), pp.1290–1327. | MR | Zbl | EuDML

[56] Redig (F.)Mathematical aspects of the abelian sand pile model, pp. 657–730 in Mathematical statistical physics, Elsevier, Amsterdam, 2006. | MR

[57] Scheidegger (A. E.)A stochastic model for drainage patterns into a intramontane trench, Bull. Assoc. Sci. Hydrol., t.12 (1967), pp.15–20.

[58] Smoluchowski (M.)Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen, Physik. Zeitschr., t.17 (1916), pp.557–599.

[59] Sornette (D.)Critical market crashes, Physics Reports, t.378 (2003), pp.1–98. | MR | Zbl

[60] Stahl (A.)Personnal communication, 2010.

[61] Stauffer (D.) & Sornette (D.)Self-organized percolation model for stock market fluctuations, Physica A, t.271 (1999), pp.496–506.

[62] Tardos (G.)Polynomial bound for a chip firing game on graphs, SIAM J. Discrete Math., t.1 (1988), pp.397–398. | MR | Zbl

[63] Velenik (Y.)Le modèle d’Ising, (2009), /cel-00392289.

[64] Volkov (S.)Forest fires on + with ignition only at 0, preprint, http://, 2009. | Zbl | MR

[65] Zhang (Y. C.)Scaling theory of self-organized criticality, Phys. Rev. Lett., t.63 (1989), pp.470–473.

[66] Zinck (R.) & Grimm (V.)More realistic than anticipated: a classical forest-fire model from statistical physics captures real fire shapes, The Open Ecology J., t.1 (2008), pp.8–13.

[67] Zinck (R.), Johst (K.) & Grimm (V.)Wildfire, landscape diversity and the Drossel-Schwabl model, Ecological Modelling, t.221 (2010), pp.98–105.

Cited by Sources: