Crossing estimates on general s-embeddings

We prove Russo-Seymour-Welsh type crossing estimates for the FK-Ising model on general s-embeddings whose origami map has an asymptotic Lipschitz constant strictly smaller than \(1\), provided a mild non-degeneracy assumption is satisfied. This result extends the original work of Chelkak and provides a general framework to prove that connection probabilities between boundaries of boxes remain bounded away from \(0\) and \(1\). It is explained that one cannot prove similar estimates without a similar assumption on the origami map, and allows to propose some notion of critical model for generic planar graphs, that can be rephrased from the perspective of the associated propagator operator. This paper reproves along the way corresponding results in almost all already known setups and also treats new ones of interest.