Timár, Adám Neighboring clusters in Bernoulli percolation. (English) Zbl 1112.60085 Ann. Probab. 34, No. 6, 2332-2343 (2006). 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)]. Cited in 4 Documents MSC: 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B43 Percolation 60B99 Probability theory on algebraic and topological structures Keywords:cluster repulsion; percolation; nonamenable; touching clusters Citations:Zbl 0948.60098 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Antal, P. and Pisztora, A. (1996). On the chemical distance in supercritical Bernoulli percolation. Ann. Probab. 24 1036–1048. · Zbl 0871.60089 · doi:10.1214/aop/1039639377 [2] Benjamini, I., Kesten, H., Peres, Y. and Schramm, O. (2004). Geometry of the uniform spanning forest: Transitions in dimensions 4, 8, 12\(,\ldots.\) Ann. of Math. ( 2 ) 160 465–491. · Zbl 1071.60006 · doi:10.4007/annals.2004.160.465 [3] Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (1999). Group-invariant percolation on graphs. Geom. Funct. Anal. 9 29–66. · Zbl 0924.43002 · doi:10.1007/s000390050080 [4] Burton, R. M. and Keane, M. (1989). Density and uniqueness in percolation. Comm. Math. Phys. 121 501–505. · Zbl 0662.60113 · doi:10.1007/BF01217735 [5] Häggström, O. (1997). Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab. 25 1423–1436. · Zbl 0895.60098 · doi:10.1214/aop/1024404518 [6] Häggström, O., Peres, Y. and Schonmann, R. (1999). Percolation on transitive graphs as a coalescent process: Relentless merging followed by simultaneous uniqueness. In Perplexing Probability Problems : Festschrift in Honor of Harry Kesten (M. Bramson and R. Durrett, eds.) 69–90. Birkhäuser, Boston. · Zbl 0948.60098 [7] Lyons, R. (2000). Phase transitions on nonamenable graphs. J. Math. Phys. 41 1099–1126. · Zbl 1034.82014 · doi:10.1063/1.533179 [8] Lyons, R. and Schramm, O. (1999). Indistinguishability of percolation clusters. Ann. Probab. 27 1809–1836. · Zbl 0960.60013 · doi:10.1214/aop/1022677549 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.