Conformal Invariance of Boundary Touching Loops of FK Ising Model

Antti Kemppainen,Stanislav Smirnov

doi:10.1007/s00220-019-03437-0

Abstract

In this article we show the convergence of a loop ensemble of interfaces in the FK Ising model at criticality, as the lattice mesh tends to zero, to a unique conformally invariant scaling limit. The discrete loop ensemble is described by a canonical tree glued from the interfaces, which then is shown to converge to a tree of branching SLEs. The loop ensemble contains unboundedly many loops and hence our result describes the joint law of infinitely many loops in terms of SLE type processes, and the result gives the full scaling limit of the FK Ising model in the sense of random geometry of the interfaces. Some other results in this article are convergence of the exploration process of the loop ensemble (or the branch of the exploration tree) to hbox {SLE}(kappa ,kappa -6), kappa =16/3, and convergence of a generalization of this process for 4 marked points to hbox {SLE}[kappa ,Z], kappa =16/3, where Z refers to a partition function. The latter SLE process is a process that can’t be written as a hbox {SLE}(kappa ,rho _1,rho _2,ldots ) process, which are the most commonly considered generalizations of SLEs.

Highlights

In this article we show the convergence of a loop ensemble of interfaces in the FK Ising model at criticality, as the lattice mesh tends to zero, to a unique conformally invariant scaling limit
The discrete loop ensemble is described by a canonical tree glued from the interfaces, which is shown to converge to a tree of branching SLEs
The loop ensemble contains unboundedly many loops and our result describes the joint law of infinitely many loops in terms of SLE type processes, and the result gives the full scaling limit of the FK Ising model in the sense of random geometry of the interfaces

Summary

Introduction

The so called loop representation of the FK model is defined on so called medial lattice, shown in Fig. 1a and formed by the corners of the squares, by taking the inner and outer boundaries of all the connected components in a configuration of edges. The (chordal) exploration tree connects the fixed root vertex to any other boundary point. The general case follows from the sequel [15] of this article on the radial exploration tree of the FK Ising model These restrictions are technical and they are not needed, for instance, for the convergence in the so-called 4-point case The mapping from the collection of boundary touching loops L∂ to the chordal exploration tree T is a bijection.

Tightness of Trees and Loop Collections

Preholomorphic Martingale Observable

Characterization of the Scaling Limit