## The Annals of Probability

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

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

Xavier Bressaud and Nicolas Fournier

#### Abstract

We consider an interacting particle system (*η*_{t})_{t≥0} with values in {0, 1}^{ℤ}, 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