Bond percolation does not simulate site percolation

TL;DR

The paper establishes that site percolation cannot be approximated by bond percolation on local neighborhoods, demonstrating this first for $K_{1,4}$ and then for $K_{1,3}$ using a novel decision-tree framework and a computer-assisted linear-programming approach. It builds on the hyperedge equivalence between site and hyperedge percolation and leverages the van den Berg–Kesten inequality to derive strong connectivity inequalities, obtaining progressively tighter bounds on the best possible approximation—culminating in $ ext{α}_3 o ext{≤ }0.369$ and an Aas-type inequality for connectivity probabilities. The work clarifies a fundamental separation between site/hyperedge and bond percolation models, providing both qualitative impossibility results and quantitative inequalities that deepen understanding of connectivity under inhomogeneous percolation. It also introduces methods that can be used to tackle related conjectures and to derive further inequalities in percolation theory.

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.

Figures (4)

• Figure 1: $T$ corresponding to the Example \ref{['ex:2edges']}
• Figure 2: $S_1$, $S_2$ and $S_3$ for the case $C_1 \in a|b|c$. Regions surrounding $a, b, c$ depict $K_a$, $K_b$ and $K_c$. Respective sets are in blue and their complements are in red.
• Figure 3: Graph for $\alpha_3$.
• Figure 4: Best $\min(\mu(abc), \mu(a|b|c))$ achieved on each training epoch.

Theorems & Definitions (45)

• Definition 1.1: Connectivity
• Definition 1.3: Exact simulation
• Remark 1.4
• Theorem 1.5
• proof
• Definition 1.6: Approximate simulation
• Theorem 1.7
• Theorem 1.8
• Theorem 2.1: Site--hyperedge equivalence
• proof