# Configurations of FK Ising interfaces and hypergeometric SLE

## TL;DR

This paper proves that FK Ising model interfaces with specific boundary conditions converge to hypergeometric SLE in the scaling limit, providing a new connection between discrete models and conformal invariants.

## Contribution

It establishes the convergence of FK Ising interfaces with certain boundary conditions to hypergeometric SLE, introducing an algorithm to sample joint interface configurations.

## Key findings

- Interfaces converge to hypergeometric SLE in the scaling limit.
- An algorithm for sampling joint interface configurations is developed.
- The joint law of interfaces can be described via conformal invariants.

## Abstract

In this paper, we show that the interfaces in FK Ising model in any domain with 4 marked boundary points and wired--free--wired--free boundary conditions conditioned on a specific internal arc configuration of interfaces converge in the scaling limit to hypergeometric SLE (hSLE). The arc configuration consists of a pair of interfaces and the scaling limit of their joint law can be described by an algorithm to sample the pair from a hSLE curve and a chordal SLE (in a random domain defined by the hSLE).