Bond percolation does not simulate site percolation

TL;DR

This paper demonstrates that site percolation cannot be effectively simulated by bond percolation, using probabilistic inequalities and decision tree techniques to establish fundamental differences between the models.

Contribution

It introduces new methods to prove the non-simulability of site percolation by bond percolation, including the application of the vdBK inequality and a decision tree approach.

Findings

Site percolation cannot be simulated by bond percolation on degree 4 neighborhoods.

The decision tree technique extends to degree 3 neighborhoods.

The methods can derive inequalities for connectedness probabilities, supporting a conjecture by Erik Aas.

Abstract

We show that a site percolation is a stronger model than a bond percolation. We use the van den Berg -- Kesten (vdBK) inequality to prove that site percolation on a neighborhood of a vertex of degree $4$ cannot be simulated even approximately by bond percolation, and develop a decision tree technique to prove the same for a neighborhood of a vertex of degree $3$. This technique can be used to obtain inequalities for connectedness probabilities, including a conjectured inequality of Erik Aas.