On multiple SLE for the FK-Ising model

TL;DR

This paper proves the convergence of multiple interfaces in the critical FK-Ising model, describing their scaling limit as multiple SLE(16/3) and connecting it to spin correlations in the critical Ising model.

Contribution

It provides the first rigorous proof of multiple SLE convergence for the FK-Ising model and links the partition function to spin correlations via explicit formulas.

Findings

Convergence of multiple interfaces to multiple SLE(16/3)

Explicit expression for the partition function matching spin correlations

Recovery of BPZ equations for spin correlations

Abstract

We prove convergence of multiple interfaces in the critical planar q = 2 random cluster model, and provide an explicit description of the scaling limit. Remarkably, the expression for the partition function of the resulting multiple SLE(16/3) coincides with the bulk spin correlation in the critical Ising model in the half-plane, after formally replacing a position of each spin and its complex conjugate with a pair of points on the real line. As a corollary, we recover Belavin-Polyakov-Zamolodchikov equations for the spin correlations.