Meteor process on $${\mathbb Z}^d$$

Abstract

The meteor process is a model for mass redistribution on a graph. The case of finite graphs was analyzed in Billey et al. (On meteors, earthworms and WIMPs. Ann Appl Probab, 2014). This paper is devoted to the meteor process on $${\mathbb {Z}}^d$$ . The process is constructed and a stationary distribution is found. Convergence to this stationary distribution is proved for a large family of initial distributions. The first two moments of the mass distribution at a vertex are computed for the stationary distribution. For the one-dimensional lattice $${\mathbb {Z}}$$ , the net flow of mass between adjacent vertices is shown to have bounded variance as time goes to infinity. An alternative representation of the process on $${\mathbb {Z}}$$ as a collection of non-crossing paths is presented. The distributions of a “tracer particle” in this system of non-crossing paths are shown to be tight as time goes to infinity.