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Bond percolation does not simulate site percolation
Nikita Gladkov, Aleksandr Zimin · 2024-04-13 · via math.PR updates on arXiv.org

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.

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