The scaling limit of critical Ising interfaces is CLE(3)

TL;DR

This paper proves that the interfaces in the critical planar Ising model with + boundary conditions converge to the conformally invariant CLE(3), using coupling with FK representation and exploration processes.

Contribution

It establishes the convergence of critical Ising interfaces to CLE(3) and introduces a recursive exploration process leveraging FK-Ising interactions.

Findings

Ising interfaces converge to CLE(3) in the scaling limit.

Development of a recursive exploration process for Ising loops.

Analysis of double points in FK-Ising interfaces.

Abstract

In this paper, we consider the set of interfaces between + and - spins arising for the critical planar Ising model on a domain with + boundary conditions, and show that it converges towards CLE(3). Our proof relies on the study of the coupling between the Ising model and its random cluster (FK) representation, and of the interactions between FK and Ising interfaces. The main idea is to construct an exploration process starting from the boundary of the domain, to discover the Ising loops and to establish its convergence to a conformally invariant limit. The challenge is that Ising loops do not touch the boundary; we use the fact that FK loops touch the boundary (and hence can be explored from the boundary) and that Ising loops in turn touch FK loops, to construct a recursive exploration process that visits all the macroscopic loops. A key ingredient in the proof is the convergence of Ising free arcs to the Free Arc Ensemble (FAE), established in [BDH16]. Qualitative estimates about the Ising interfaces then allow one to identify the scaling limit of Ising loops as a conformally invariant collection of simple, disjoint SLE(3)-like loops and thus by the Markovian characterization of [ShWe12] as a CLE(3). A technical point of independent interest contained in this paper is an investigation of double points of interfaces in the scaling limit of critical FK-Ising. It relies on the technology of [KeSm12].