### Abstract

For a locally finite, connected graph G with distinguished vertex 0, let p_{c}(G) be the usual critical probability for bond percolation on G, and p_{cut}(G) = sup(p: inf_{Π} E_{p}|C(0)⋂ Π| = 0) (≤p_{c}), where Π ranges over cutsets (sets of vertices “separating 0 from ∞”), E_{p} refers to (Bernoulli bond) percolation with p the probability that an edge is open, and C(0) is the open cluster containing 0.(The definition is easily seen to be independent of the choice of distinguished vertex.) We disprove a conjecture of Russ Lyons stating that p_{cut}(G) = p_{c}(G) for every G, and propose a possible alternative.

Original language | English (US) |
---|---|

Pages (from-to) | 184-187 |

Number of pages | 4 |

Journal | Electronic Communications in Probability |

Volume | 8 |

DOIs | |

State | Published - Jan 1 2003 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Keywords

- Critical probability
- Percolation

### Cite this

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**Inequality of two critical probabilities for percolation.** / Kahn, Jeffry.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Inequality of two critical probabilities for percolation

AU - Kahn, Jeffry

PY - 2003/1/1

Y1 - 2003/1/1

N2 - For a locally finite, connected graph G with distinguished vertex 0, let pc(G) be the usual critical probability for bond percolation on G, and pcut(G) = sup(p: infΠ Ep|C(0)⋂ Π| = 0) (≤pc), where Π ranges over cutsets (sets of vertices “separating 0 from ∞”), Ep refers to (Bernoulli bond) percolation with p the probability that an edge is open, and C(0) is the open cluster containing 0.(The definition is easily seen to be independent of the choice of distinguished vertex.) We disprove a conjecture of Russ Lyons stating that pcut(G) = pc(G) for every G, and propose a possible alternative.

AB - For a locally finite, connected graph G with distinguished vertex 0, let pc(G) be the usual critical probability for bond percolation on G, and pcut(G) = sup(p: infΠ Ep|C(0)⋂ Π| = 0) (≤pc), where Π ranges over cutsets (sets of vertices “separating 0 from ∞”), Ep refers to (Bernoulli bond) percolation with p the probability that an edge is open, and C(0) is the open cluster containing 0.(The definition is easily seen to be independent of the choice of distinguished vertex.) We disprove a conjecture of Russ Lyons stating that pcut(G) = pc(G) for every G, and propose a possible alternative.

KW - Critical probability

KW - Percolation

UR - http://www.scopus.com/inward/record.url?scp=3042521545&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=3042521545&partnerID=8YFLogxK

U2 - 10.1214/ECP.v8-1099

DO - 10.1214/ECP.v8-1099

M3 - Article

AN - SCOPUS:3042521545

VL - 8

SP - 184

EP - 187

JO - Electronic Communications in Probability

JF - Electronic Communications in Probability

SN - 1083-589X

ER -