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

TL;DR

This paper extends the meteor process to the infinite lattice ${\\mathbb Z}^d$, establishing its stationary distribution, convergence properties, and analyzing mass flow and tracer particle behavior.

Contribution

It constructs the meteor process on ${\mathbb Z}^d$, finds its stationary distribution, and analyzes convergence and flow properties, including a novel path representation for ${\mathbb Z}$.

Findings

Stationary distribution exists and is characterized.

Convergence to stationarity is proved for various initial conditions.

Mass flow variance remains bounded over time in ${\mathbb Z}$.

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.