Strong RSW estimates for the FK-Ising model on s-embeddings

Abstract: We adapt the 'induction over scales' scheme of Duminil-Copin, Manolescu, and Tassion to the FK-Ising model on s-embeddings. Namely, we prove the existence of crossings of rough topological quadrilaterals with unfavorable boundary conditions, known as 'strong' RSW, under the 'uniformly bounded geometry' assumption (all edge lengths are comparable to 1 and all angles are bounded from below) on the tangential quads that form an s-embedding. This result has a number of standard consequences like quasi-multiplicativity of arm events and fractal properties of the clusters. It can be applied in a variety of setups including near-critical Ising models on regular grids below the correlation length scale by considering appropriate s-embedding thereof.