On considère la dynamique aléatoire suivante sur un arbre de Cayley uniforme avec sommets et pour lequel les arêtes peuvent être inflammables, ignifugées, ou brûlées. Au temps initial, toutes les arêtes sont inflammables, et chaque arête inflammable est remplacée à taux par une arête ignifugée, indépendamment des autres arêtes. Par ailleurs, une arête inflammable peut également prendre feu avec un taux , et le feu se propage alors le long des arêtes inflammables voisines et n’est stoppé que par les arêtes ignifugées. Nous montrons que lorsque , la densité terminale des sommets ignifugés converge vers si , vers si , et vers une variable aléatoire non dégénérée pour . On étudie ensuite la connectivité de la forêt ignifugée, et plus particulièrement l’existence de composantes géantes.
We consider random dynamics on the edges of a uniform Cayley tree with vertices, in which edges are either flammable, fireproof, or burnt. Every flammable edge is replaced by a fireproof edge at unit rate, while fires start at smaller rate on each flammable edge, then propagate through the neighboring flammable edges and are only stopped at fireproof edges. A vertex is called fireproof when all its adjacent edges are fireproof. We show that as , the terminal density of fireproof vertices converges to when , to when , and to some non-degenerate random variable when . We further study the connectivity of the fireproof forest, in particular the existence of a giant component.
@article{AIHPB_2012__48_4_909_0, author = {Bertoin, Jean}, title = {Fires on trees}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, volume = {48}, year = {2012}, pages = {909-921}, doi = {10.1214/11-AIHP435}, mrnumber = {3052398}, zbl = {1263.60083}, language = {en}, url = {http://dml.mathdoc.fr/item/AIHPB_2012__48_4_909_0} }
Bertoin, Jean. Fires on trees. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) pp. 909-921. doi : 10.1214/11-AIHP435. http://gdmltest.u-ga.fr/item/AIHPB_2012__48_4_909_0/
[1] The continuum random tree III. Ann. Probab. 21 (1993) 248-289. | MR 1207226 | Zbl 0791.60009
.[2] Tree-valued Markov chains derived from Galton-Watson processes. Ann. Inst. H. Poincaré Probab. Stat. 34 (1998) 637-686. | Numdam | MR 1641670 | Zbl 0917.60082
and .[3] The standard additive coalescent. Ann. Probab. 26 (1998) 1703-1726. | MR 1675063 | Zbl 0936.60064
and .[4] Self-similar fragmentations. Ann. Inst. H. Poincaré Probab. Stat. 38 (2002) 319-340. | Numdam | MR 1899456 | Zbl 1002.60072
.[5] Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2006. | MR 2253162 | Zbl 1107.60002
.[6] Self-organized critical forest fire model. Phys. Rev. Lett. 69 (1992) 1629-1632.
and .[7] The genealogy of self-similar fragmentations with negative index as a continuum random tree. Electron. J. Probab. 9 (2004) 57-97 (electronic). | MR 2041829 | Zbl 1064.60076
and .[8] Scaling limits of Markov branching trees, with applications to Galton-Watson and random unordered trees. Ann. Probab. To appear. Available at http://arxiv.org/abs/1003.3632. | MR 3050512 | Zbl 1259.60033
and .[9] Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models. Ann. Probab. 36 (2008) 1790-1837. | MR 2440924 | Zbl 1155.92033
, , and .[10] Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen, 1971. | MR 322926 | Zbl 0219.60027
and .[11] Random cutting and records in deterministic and random trees. Random Structures Algorithms 29 (2006) 139-179. | MR 2245498 | Zbl 1120.05083
.[12] Critical mean field frozen percolation and the multiplicative coalescent with linear deletion. In preparation.
and .[13] Cutting down random trees. J. Australian Math. Soc. 11 (1970) 313-324. | MR 284370 | Zbl 0196.27602
and .[14] Cutting down very simple trees. Quaest. Math. 29 (2006) 211-227. | MR 2233368 | Zbl 1120.05020
.[15] The asymptotic distribution of maximum tree size in a random forest. Theor. Probab. Appl. 22 (1977) 509-520. | MR 461619 | Zbl 0386.60011
.[16] Coalescent random forests. J. Combin. Theory Ser. A 85 (1999) 165-193. | MR 1673928 | Zbl 0918.05042
.[17] Mean field frozen percolation. J. Stat. Phys. 137 (2009) 459-499. | MR 2564286 | Zbl 1181.82034
.