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 \cite{BBPS}. 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.