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.
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Statistics, Probability and Uncertainty
- Critical probability