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 $\{0,1\}^{\mathbb{Z}}$, 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.