zbMATH — the first resource for mathematics

On the invariant distribution of a one-dimensional avalanche process. (English) Zbl 1171.60022
The authors consider an interacting particle system derived by an independent family of Poisson processes with rate 1. It is shown that such a process has a unique stationary distribution, which is exponentially mixing and can be simulated. In addition, authors prove that the process tends to equilibrium exponentially fast as time tends to infinity. Finally, it is shown numerically that the invariant distribution of a related mean-field coagulation-fragmentation model is very close to that of the interacting particle system.

60K35 Interacting random processes; statistical mechanics type models; percolation theory
Full Text: DOI arXiv
[1] Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists. Bernoulli 5 3-48. · Zbl 0930.60096 · doi:10.2307/3318611
[2] Bak, P., Tang, C. and Wiesenfeld, K. (1987). Self-organized criticality: An explanation of 1/f noise. Phys. Rev. Lett. 59 381-384. · Zbl 1230.37103 · doi:10.1103/PhysRevA.38.364
[3] Bak, P., Tang, C. and Wiesenfeld, K. (1988). Self-organized criticality. Phys. Rev. A (3) 38 364-374. · Zbl 1230.37103 · doi:10.1103/PhysRevA.38.364
[4] van den Berg, J. and Brouwer, R. (2006). Self-organized forest-fires near the critical time. Comm. Math. Phys. 267 265-277. · Zbl 1111.60080 · doi:10.1007/s00220-006-0025-1
[5] van den Berg, J. and Járai, A. A. (2005). On the asymptotic density in a one-dimensional self-organized critical forest-fire model. Comm. Math. Phys. 253 633-644. · Zbl 1079.82010 · doi:10.1007/s00220-004-1200-x
[6] Brouwer, R. and Pennanen, J. (2006). The cluster size distribution for a forest-fire process on \(. Electron. J. Probab. 11 1133-1143.\) · Zbl 1128.60080 · eudml:128292
[7] Dhar, D. (2006). Theoretical studies of self-organized criticality. Phys. A 369 29-70. · doi:10.1016/j.physa.2006.04.004
[8] Drossel, B. and Schwabl, F. (1992). Self-organized critical forest-fire model. Phys. Rev. Lett. 69 1629-1632.
[9] Dürre, M. (2006). Existence of multi-dimensional infinite volume self-organized critical forest-fire models. Electron. J. Probab. 11 513-539. · Zbl 1109.60081 · eudml:127184
[10] Dürre, M. (2006). Uniqueness of multi-dimensional infinite volume self-organized critical forest-fire models. Electron. Comm. Probab. 11 304-315. · Zbl 1130.60091 · eudml:128418
[11] Fournier, N. and Mischler, S. (2004). Exponential trend to equilibrium for discrete coagulation equations with strong fragmentation and without a balance condition. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 2477-2486. JSTOR: · Zbl 1170.82349 · doi:10.1098/rspa.2004.1294 · links.jstor.org
[12] Grassberger, P. (2002). Critical behaviour of the Drossel-Schwabl forest fire model. New J. Phys. 4 . 17.
[13] Henley, C. L. (1989). Self-organized percolation: A simpler model. Bull. Amer. Phys. Soc. 34 838.
[14] Laurençot, P. and Mischler, S. (2004). On coalescence equations and related models. In Modeling and Computational Methods for Kinetic Equations. Model. Simul. Sci. Eng. Technol. 321-356. Birkhäuser, Boston. · Zbl 1105.82027
[15] Liggett, T. M. (1985). Interacting Particle Systems . Springer, New York. · Zbl 0559.60078
[16] Olami, Z., Feder, H. J. S. and Christensen, K. (1992). Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes. Phys. Rev. Lett. 68 1244-1247.
[17] Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. In Proceedings of the Seventh International Conference on Random Structures and Algorithms ( Atlanta, GA , 1995) 9 223-252. · Zbl 0859.60067 · doi:10.1002/(SICI)1098-2418(199608/09)9:1/2<223::AID-RSA14>3.0.CO;2-O
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.