Connection probabilities of multiple FK-Ising interfaces

Abstract

AbstractWe find the scaling limits of a general class of boundary-to-boundary connection probabilities and multiple interfaces in the critical planar FK-Ising model, thus verifying predictions from the physics literature. We also discuss conjectural formulas using Coulomb gas integrals for the corresponding quantities in general critical planar random-cluster models with cluster-weight $${q \in [1,4)}$$ q ∈ [ 1 , 4 ) . Thus far, proofs for convergence, including ours, rely on discrete complex analysis techniques and are beyond reach for other values of q than the FK-Ising model ($$q=2$$ q = 2 ). Given the convergence of interfaces, the conjectural formulas for other values of q could be verified similarly with relatively minor technical work. The limit interfaces are variants of $$\text {SLE}_\kappa $$ SLE κ curves (with $$\kappa = 16/3$$ κ = 16 / 3 for $$q=2$$ q = 2 ). Their partition functions, that give the connection probabilities, also satisfy properties predicted for correlation functions in conformal field theory (CFT), expected to describe scaling limits of critical random-cluster models. We verify these properties for all $$q \in [1,4)$$ q ∈ [ 1 , 4 ) , thus providing further evidence of the expected CFT description of these models.