Positive dependence for colored percolation

TL;DR

This paper investigates the dependence structure of colored percolation models, establishing positive and negative dependence results for certain color pairs, and applies these findings to crossing probabilities and critical percolation scenarios.

Contribution

It introduces a generalized Harris--Kleitman inequality to analyze dependence in 4-color edge percolation models, providing new insights into their probabilistic behavior.

Findings

Positive dependence for pairs ab, ac

Negative dependence for pairs ab, ac and bc

Applications to crossing probabilities and colored critical 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 joint positive dependence for pairs of colors ab, ac and ac, and joint negative 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.