Electronic Communications in Probability

Inequality of Two Critical Probabilities for Percolation

Jeff Kahn

Full-text: Open access

Abstract

We disprove a conjecture of Russ Lyons---that for every locally finite, connected graph $G$, the critical probability for (Bernoulli bond) percolation on $G$ is equal to the "first moment method" lower bound on this probability---and propose a possible alternative.

Article information

Source
Electron. Commun. Probab., Volume 8 (2003), paper no. 21, 184-187.

Dates
Accepted: 27 December 2003
First available in Project Euclid: 18 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1463608905

Digital Object Identifier
doi:10.1214/ECP.v8-1099

Mathematical Reviews number (MathSciNet)
MR2042758

Zentralblatt MATH identifier
1060.60096

Subjects
Primary: 60C05: Combinatorial probability 82B43: Percolation [See also 60K35]

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Kahn, Jeff. Inequality of Two Critical Probabilities for Percolation. Electron. Commun. Probab. 8 (2003), paper no. 21, 184--187. doi:10.1214/ECP.v8-1099. https://projecteuclid.org/euclid.ecp/1463608905


Export citation

References

  • M. Aizenman and D.J. Barsky, Sharpness of the phase transition in percolation models, Comm. Math. Phys. 108 (1987), 489-526.
  • G. Grimmett, Percolation, (Second edition), Springer-Verlag, Berlin, 1999.
  • R. Lyons, Random walks and percolation on trees, Ann. Probab. 18 (1990), 931-958.
  • R. Lyons, The Ising model and percolation on trees and tree-like graphs, Comm. Math. Phys. 125 (1989), 337-353.
  • R. Lyons and Y. Peres, Probability on Trees and Networks. URL: http://mypage.iu.edu/~rdlyons/#book
  • M.V. Menshikov, S.A. Molchanov and S.A. Sidorenko, Percolation theory and some applications, J. Soviet Math. 42 (1988), 1766-1810 (translated from Itogi Nauki i Tekhniki 24 (1986), 53-110).