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Escape of resources in a distributed clustering process. (English) Zbl 1226.60132

Summary: In a distributed clustering algorithm introduced by E. G. Coffman jun., P.-J. Courtois, E. N. Gilbert and P. Piret [J. Appl. Probab. 28, No. 4, 737–750 (1991; Zbl 0741.60114)], 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 [M. R. Hilário et al., Commun. Pure Appl. Math. 63, No. 7, 926–934 (2010; Zbl 1202.60156)], 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 [J. van den Berg and R. W. J. Meester, Random Struct. Algorithms 2, No. 3, 335–341 (1991; Zbl 0741.05072)].
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 [Hilário et al., loc. cit.].

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
68M14 Distributed systems
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