The Annals of Probability

On the invariant distribution of a one-dimensional avalanche process

Xavier Bressaud and Nicolas Fournier

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Abstract

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.

Article information

Source
Ann. Probab., Volume 37, Number 1 (2009), 48-77.

Dates
First available in Project Euclid: 17 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1234881684

Digital Object Identifier
doi:10.1214/08-AOP396

Mathematical Reviews number (MathSciNet)
MR2489159

Zentralblatt MATH identifier
1171.60022

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Stochastic interacting particle systems equilibrium coalescence fragmentation self-organized criticality forest-fire model

Citation

Bressaud, Xavier; Fournier, Nicolas. On the invariant distribution of a one-dimensional avalanche process. Ann. Probab. 37 (2009), no. 1, 48--77. doi:10.1214/08-AOP396. https://projecteuclid.org/euclid.aop/1234881684


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