Percolation probabilities on the square lattice

P. D. Seymour, D. J.A. Welsh

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130 Scopus citations


This chapter focuses on bond percolation on the square lattice and briefly describes percolation model; this model is a special but perhaps the most interesting case of the general theory of percolation. It introduces the FKG inequality of Fortuin, Kasteleyn, and Ginibre; it proves a remarkable inequality showing that nondecreasing functions on a finite distributive lattice are positively correlated by all positive measures, which have a certain convexity property. The problem of percolation through an n × n sponge is introduced. The chapter also examines two of the possible critical probabilities pT and pH and uses the theory developed for the sponge problem to prove the result pT + pH = 1. By the percolation model on G, one mean the assignment of open or closed to each edge of G with probabilities p and q = 1 - p respectively, the assignments to be independent for each edge.

Original languageEnglish (US)
Pages (from-to)227-245
Number of pages19
JournalAnnals of Discrete Mathematics
Issue numberC
StatePublished - Jan 1 1978
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics


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