Inequality of two critical probabilities for percolation

Jeff Kahn

doi:10.1214/ECP.v8-1099

Inequality of two critical probabilities for percolation

Jeff Kahn

School of Arts and Sciences, Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

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	https://doi.org/10.1214/ECP.v8-1099
State	Published - Jan 1 2003

All Science Journal Classification (ASJC) codes

Statistics and Probability
Statistics, Probability and Uncertainty

Keywords

Critical probability
Percolation

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10.1214/ECP.v8-1099

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abstract = "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.",

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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

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JO - Electronic Communications in Probability

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ER -

Inequality of two critical probabilities for percolation

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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