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January 2009 On the invariant distribution of a one-dimensional avalanche process
Xavier Bressaud, Nicolas Fournier
Ann. Probab. 37(1): 48-77 (January 2009). DOI: 10.1214/08-AOP396


We consider an interacting particle system (ηt)t≥0 with values in {0, 1}, in which each vacant site becomes occupied with rate 1, while each connected component of occupied sites become vacant with rate equal to its size. We show that such a process admits a unique invariant distribution, which is exponentially mixing and can be perfectly simulated. We also prove that for any initial condition, the avalanche process tends to equilibrium exponentially fast, as time increases to infinity. Finally, we consider a related mean-field coagulation–fragmentation model, we compute its invariant distribution and we show numerically that it is very close to that of the interacting particle system.


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Xavier Bressaud. Nicolas Fournier. "On the invariant distribution of a one-dimensional avalanche process." Ann. Probab. 37 (1) 48 - 77, January 2009.


Published: January 2009
First available in Project Euclid: 17 February 2009

zbMATH: 1171.60022
MathSciNet: MR2489159
Digital Object Identifier: 10.1214/08-AOP396

Primary: 60K35

Keywords: Coalescence , Equilibrium , forest-fire model , fragmentation , Self-organized criticality , stochastic interacting particle systems

Rights: Copyright © 2009 Institute of Mathematical Statistics


Vol.37 • No. 1 • January 2009
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