The Annals of Probability

Critical Percolation on Any Nonamenable Group has no Infinite Clusters

Itai Benjamini, Russell Lyons, Yuval Peres, and Oded Schramm

Full-text: Open access

Abstract

We show that independent percolation on any Cayley graph of a nonamenable group has no infinite components at the critical parameter. This result was obtained by the present authors earlier as a corollary of a general study of group-invariant percolation. The goal here is to present a simpler self-contained proof that easily extends to quasi-transitive graphs with a unimodular automorphism group. The key tool is a “mass-transport” method, which is a technique of averaging in nonamenable settings.

Article information

Source
Ann. Probab., Volume 27, Number 3 (1999), 1347-1356.

Dates
First available in Project Euclid: 29 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022677450

Digital Object Identifier
doi:10.1214/aop/1022677450

Mathematical Reviews number (MathSciNet)
MR1733151

Zentralblatt MATH identifier
0961.60015

Subjects
Primary: 60B99: None of the above, but in this section
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 82B43

Keywords
Percolation Cayley graphs amenability

Citation

Benjamini, Itai; Lyons, Russell; Peres, Yuval; Schramm, Oded. Critical Percolation on Any Nonamenable Group has no Infinite Clusters. Ann. Probab. 27 (1999), no. 3, 1347--1356. doi:10.1214/aop/1022677450. https://projecteuclid.org/euclid.aop/1022677450


Export citation

References

  • ADAMS, S. 1990. Trees and amenable equivalence relations. Ergodic Theory Dynamic Systems 10 1 14.
  • AIZENMAN, M., KESTEN, H. and NEWMAN, C. M. 1987. Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Comm. Math. Phys. 111 505 531.
  • BABSON, E. and BENJAMINI, I. 1999. Cut sets and normed cohomology with applications to percolation. Proc. Amer. Math. Soc. 127 589 597.
  • BENJAMINI, I., LYONS, R., PERES, Y. and SCHRAMM, O. 1999. Group-invariant percolation on graphs. Geom. Funct. Anal. 9 29 66.d
  • BENJAMINI, I. and SCHRAMM, O. 1996. Percolation beyond, many questions and a few answers. Electronic Comm. Probab. 1 71 82.
  • BENJAMINI, I. and SCHRAMM, O. 1998a. Percolation in the hyperbolic plane. Unpublished manuscript.d
  • BENJAMINI, I. and SCHRAMM, O. 1998b. Recent progress on percolation beyond. http: www.wisdom.weizmann.ac.il schramm papers pyond-rep.
  • BURTON, R. M. and KEANE, M. 1989. Density and uniqueness in percolation. Comm. Math. Phys. 121 501 505.
  • GRIMMETT, G. R. 1989. Percolation. Springer, New York.
  • GRIMMETT, G. R. 1995. The stochastic random-cluster process and the uniqueness of randomcluster measures. Ann. Probab. 23 1461 1510.
  • GRIMMETT, G. R. and NEWMAN, C. M. 1990. Percolation in 1 dimensions. In Disorder inPhysical Systems G. R. Grimmett and D. J. A. Welsh, eds. 219 240. Clarendon Press, Oxford.Z.
  • HAGGSTROM, O. 1996. The random-cluster model on a homogeneous tree. Probab. Theory ¨ ¨ Related Fields 104 231 253.
  • HAGGSTROM, O. 1997. Infinite clusters in dependent automorphism invariant percolation on ¨ ¨ trees. Ann. Probab. 25 1423 1436.
  • HAGGSTROM, O. and PERES, Y. 1998. Monotonicity of uniqueness for percolation on transitive ¨ ¨ graphs: all infinite clusters are born simultaneously. Probab. Theory Related Fields 113 273 285.
  • HAGGSTROM, O., PERES, Y. and SCHONMANN, R. 1998. Percolation on transitive graphs as a ¨ ¨ coalescent process: relentless merging followed by simultaneous uniqueness. In PerZ plexing Probability Problems: Papers in Honor of H. Kesten M. Bramson and R.. Durrett, eds.. Birkhauser, Boston. To appear. ¨
  • HARA, T. and SLADE, G. 1994. Mean-field behaviour and the lace expansion. In Probability andPhase Transition G. R. Grimmett, ed. 87 122. Kluwer, Dordrecht. Z.
  • HARRIS, T. E. 1960. A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc. 56 13 20. 1
  • KESTEN, H. 1980. The critical probability of bond percolation on the square lattice equals. 2 Comm. Math. Phys. 74 41 59.
  • LALLEY, S. P. 1996. Percolation on Fuchsian groups. Ann. Inst. H. Poincare Probab. Statist. 34 ´ 151 178.
  • LYONS, R. and SCHRAMM, O. 1998. Indistinguishability of percolation clusters. Ann. Probab. To appear.
  • NEWMAN, C. M. and SCHULMAN, L. S. 1981. Infinite clusters in percolation models. J. Statist. Phys. 26 613 628.
  • SCHONMANN, R. H. 1998. Percolation in 1 dimensions at the uniqueness threshold. In Z Perplexing Probability Problems: Papers in Honor of H. Kesten M. Bramson and R.. Durrett, eds.. Birkhauser, Boston. To appear. ¨
  • VAN DEN BERG, J. and MEESTER, R. W. J. 1991. Stability properties of a flow process in graphs. Random Structures Algorithms 2 335 341.
  • WU, C. C. 1993. Critical behavior of percolation and Markov fields on branching planes. J. Appl. Probab. 30 538 547.
  • GIVAT RAM, JERUSALEM 91904 REHOVOT 76100 ISRAEL ISRAEL AND E-MAIL: schramm@wisdom.weizmann.ac.il DEPARTMENT OF STATISTICS UNIVERSITY OF CALIFORNIA
  • BERKELEY, CALIFORNIA 94720-3860 E-MAIL: peres@stat.berkeley.edu