## Electronic Communications in Probability

### Escape of resources in a distributed clustering process

#### Abstract

In a distributed clustering algorithm introduced by Coffman, Courtois, Gilbert and Piret [1], 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 [4] 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 [2]. 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 [4].

#### Article information

Source
Electron. Commun. Probab., Volume 15 (2010), paper no. 40, 442-448.

Dates
Accepted: 30 September 2010
First available in Project Euclid: 6 June 2016

https://projecteuclid.org/euclid.ecp/1465243983

Digital Object Identifier
doi:10.1214/ECP.v15-1567

Mathematical Reviews number (MathSciNet)
MR2726090

Zentralblatt MATH identifier
1226.60132

Rights

#### Citation

van den Berg, Jacob; Hilário, Marcelo; Holroyd, Alexander. Escape of resources in a distributed clustering process. Electron. Commun. Probab. 15 (2010), paper no. 40, 442--448. doi:10.1214/ECP.v15-1567. https://projecteuclid.org/euclid.ecp/1465243983

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