L. Köhler-Schindler: Russo–Seymour–Welsh theory for FKG percolation

Abstract: We prove that the Russo–Seymour–Welsh theory is valid for any invariant bond percolation measure on \mathbb{Z}^2 satisfying the FKG inequality. This means that the crossing probability of a long rectangle is related by a universal homeomorphism to the crossing probability of a short rectangle.

The first such result was proven for Bernoulli percolation in 1978 by Russo and independently by Seymour and Welsh. It has since become one of the most important tools in the study of planar percolation. While the theory has later been extended to more general models, previous proofs relied crucially on a mixing assumption or led to substantially weaker results.

This is joint work with Vincent Tassion.


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