Positive dependence for colored percolation.

Abstract

For uniform random 4-colorings of graph edges with colors {a,b,c,d}, every two colors form a 1/2- percolation, and every two overlapping pairs of colors form independent 1/2-percolations. We show positive mutual dependence for pairs of colors ab,ac, and ad, and negative mutual dependence for pairs of colors ab,ac, and bc. The proof is based on a generalization of the Harris-Kleitman inequalities. We apply the results to crossing probabilities for the colored bond and site percolation, and to colored critical percolation that we also define.