# On the invariant distribution of a one-dimensional avalanche process

Abstract : We consider an interacting particle system $(η_t )_{t\geq 0}$ with values in ${0, 1}^Z$ , in which each vacant site becomes occupied with rate 1, while each con- nected component of occupied sites become vacant with rate equal to its size. We show that such a process admits a unique invariant distribu- tion, 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 com- pute its invariant distribution, and we show numerically that it is very close to that of the interacting particle system.
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Journal articles
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https://hal.archives-ouvertes.fr/hal-00731532
Contributor : Nicolas Fournier Connect in order to contact the contributor
Submitted on : Thursday, September 13, 2012 - 9:43:26 AM
Last modification on : Thursday, September 29, 2022 - 2:21:15 PM

### Citation

Xavier Bressaud, Nicolas Fournier. On the invariant distribution of a one-dimensional avalanche process. Annals of Probability, Institute of Mathematical Statistics, 2009, 37 (1), pp.48-77. ⟨10.1214/08-AOP396⟩. ⟨hal-00731532⟩

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