Percolation Inequalities and Decision Trees

TL;DR

This paper extends key percolation inequalities to decision trees, enabling improved estimates of connection probabilities in Bernoulli bond percolation on various graphs.

Contribution

It introduces decision tree generalizations of classical inequalities like HK and vdBK, broadening their applicability in percolation theory.

Findings

Generalized inequalities for decision trees

Enhanced bounds for connection probabilities in percolation

Applications to complex graph structures

Abstract

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.