Inequality of two critical probabilities for percolation

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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.

Original languageEnglish (US)
Pages (from-to)184-187
Number of pages4
JournalElectronic Communications in Probability
Volume8
DOIs
StatePublished - Jan 1 2003

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Critical probability
  • Percolation

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