Hilário, M. R.; Louidor, O.; Newman, C. M.; Rolla, L. T.; Sheffield, S.; Sidoravicius, V. Fixation for distributed clustering processes. (English) Zbl 1202.60156 Commun. Pure Appl. Math. 63, No. 7, 926-934 (2010). The authors consider the following process on the vertices of \(\mathbb Z^d\), where \(d \geq 1\). Assume that initially every vertex \(x\) has a load \(C_0(x)\). At each round each node transfers its load to the neighbouring node with the maximal load. The authors show that for any initial translation-invariant distribution of the load, at each vertex the process almost surely stops after finitely many steps. Their proof also applies to other lattices such as Cayley graphs and infinite regular trees, with a slight modification. Reviewer: Nikolaos Fountoulakis (Saarbrücken) Cited in 1 Review MSC: 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60D05 Geometric probability and stochastic geometry 05C90 Applications of graph theory PDFBibTeX XMLCite \textit{M. R. Hilário} et al., Commun. Pure Appl. Math. 63, No. 7, 926--934 (2010; Zbl 1202.60156) Full Text: DOI arXiv References: [1] van den Berg, On a distributed clustering process of Coffman, Courtois, Gilbert and Piret, J. Appl. Probab. 35 pp 919– (1998) · Zbl 0931.60084 [2] van den Berg, J.; Hilário, M. R.; Holroyd, A. In preparation. [3] van den Berg, Stability properties of a flow process in graphs, Random Structures Algorithms 2 pp 335– (1991) · Zbl 0741.05072 [4] Coffman, A distributed clustering process, J. Appl. Probab. 28 pp 737– (1991) · Zbl 0741.60114 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.