Percolation Inequalities and Decision Trees

The use of decision trees for percolation inequalities started with the celebrated O’Donnell–Saks–Schramm–Servedio (OSSS) inequality. We prove decision tree generalizations of the Harris–Kleitman (HK), van den Berg–Kesten (vdBK), and other inequalities. These inequalities are then applied to estimate the connection probabilities in Bernoulli bond percolation on general graphs.