Neighboring clusters in Bernoulli percolation. (English) Zbl 1112.60085

Summary: We consider Bernoulli percolation on a locally finite quasi-transitive unimodular graph and prove that two infinite clusters cannot have infinitely many pairs of vertices at distance 1 from one another or, in other words, that such graphs exhibit “cluster repulsion”. This partially answers a question of O. Häggström, Y. Peres and R. H. Schonmann [in: Perplexing problems in probability. Prog. Probab. 44, 69–90 (1999; Zbl 0948.60098)].


60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
60B99 Probability theory on algebraic and topological structures


Zbl 0948.60098
Full Text: DOI arXiv


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