Connection Probabilities of Multiple FK-Ising Interfaces

TL;DR

This paper establishes the scaling limits of connection probabilities and multiple interfaces in the critical FK-Ising model, confirming predictions from physics and conformal field theory, and discusses conjectural formulas for related models.

Contribution

It proves convergence of interfaces in the FK-Ising model and verifies properties of their connection probabilities, supporting the conformal field theory predictions.

Findings

Scaling limits of connection probabilities are established.

Interfaces converge to variants of SLE curves with specific parameters.

Partition functions match properties predicted by conformal field theory.

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.