Connection Probabilities of Multiple FK-Ising Interfaces

Abstract

We 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 $q\in [1,4)$. 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 ($q=2$). 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 ${\mathrm{SLE}}_{\kappa }$ curves (with $\kappa =16/3$ for $q=2$). 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 $q\in [1,4)$, thus providing further evidence of the expected CFT description of these models.

v3: minor fixes; final version to appear in Probab. Theory Related Fields

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Cited by in corpus (2)

• Uniform Spanning Tree in Topological Polygons, Partition Functions for SLE(8), and Correlations in $c=-2$ Logarithmic CFT
• Planar UST Branches and $c=-2$ Degenerate Boundary Correlations