Escape of resources in distributed clustering processes

Abstract

In a distributed clustering algorithm introduced by Coffman, Courtois, Gilbert and Piret \cite{coffman91}, each vertex of $\mathbb{Z}^d$ receives an initial amount of a resource, and, at each iteration, transfers all of its resource to the neighboring vertex which currently holds the maximum amount of resource. In \cite{hlrnss} it was shown that, if the distribution of the initial quantities of resource is invariant under lattice translations, then the flow of resource at each vertex eventually stops almost surely, thus solving a problem posed in \cite{berg91}. In this article we prove the existence of translation-invariant initial distributions for which resources nevertheless escape to infinity, in the sense that the the final amount of resource at a given vertex is strictly smaller in expectation than the initial amount. This answers a question posed in \cite{hlrnss}.