Percolation probabilities on the square lattice

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 p_T and p_H and uses the theory developed for the sponge problem to prove the result p_T + p_H = 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.