On critical probabilities in percolation theory

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Abstract
Several versions of the concept of critical percolation probability are discussed in the bond percolation problem on the square lattice. Critical probabilities are also employed as technical devices in the proofs of two new results. First, there is a critical probability pT below which all moments of the cluster size are finite. Secondly, an infinite connected cluster of open bonds exists with positive probability if and only if any angular sector contains an infinite connected cluster of open bonds with positive probability. An expression is derived for the expected number of open clusters per bond in the percolation model, relating to the problem of rigorously justifying a critical probability result of Sykes and Essam.

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