The Annals of Probability

Neighboring clusters in Bernoulli percolation

Adám Timár

Full-text: Open access


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 Häggström, Peres and Schonmann.

Article information

Ann. Probab., Volume 34, Number 6 (2006), 2332-2343.

First available in Project Euclid: 13 February 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]
Secondary: 60B99: None of the above, but in this section

Cluster repulsion percolation nonamenable touching clusters


Timár, Adám. Neighboring clusters in Bernoulli percolation. Ann. Probab. 34 (2006), no. 6, 2332--2343. doi:10.1214/009117906000000485.

Export citation


  • Antal, P. and Pisztora, A. (1996). On the chemical distance in supercritical Bernoulli percolation. Ann. Probab. 24 1036–1048.
  • 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.
  • Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (1999). Group-invariant percolation on graphs. Geom. Funct. Anal. 9 29–66.
  • Burton, R. M. and Keane, M. (1989). Density and uniqueness in percolation. Comm. Math. Phys. 121 501–505.
  • Häggstr öm, O. (1997). Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab. 25 1423–1436.
  • 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.
  • Lyons, R. (2000). Phase transitions on nonamenable graphs. J. Math. Phys. 41 1099–1126.
  • Lyons, R. and Schramm, O. (1999). Indistinguishability of percolation clusters. Ann. Probab. 27 1809–1836.