Conformal Invariance of Boundary Touching Loops of FK Ising Model

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 SLE(κ,κ−6), κ=16/3, and convergence of a generalization of this process for 4 marked points to SLE[κ,Z], κ=16/3, where Z refers to a partition function. The latter SLE process is a process that can't be written as a SLE(κ,ρ1,ρ2,…) process, which are the most commonly considered generalizations of SLEs.