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$(\kappa ,\kappa -6)$, $\kappa =16/3$, and convergence of a generalization of this process for $4$ marked points to 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 SLE$(\kappa ,{\rho }_1,{\rho }_2,\dots )$ process, which are the most commonly considered generalizations of SLEs.

49 pages, 13 figures; minor changes; preprint of the published version

References in corpus (6)

• Two-Dimensional Critical Percolation: The Full Scaling Limit
• Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model
• Critical percolation in the plane
• The scaling limit of critical Ising interfaces is CLE(3)
• Conformal invariance in random cluster models. II. Full scaling limit as a branching SLE
• Conformal invariance of crossing probabilities for the Ising model with free boundary conditions

Cited by in corpus (13)

• CLE percolations
• The scaling limit of critical Ising interfaces is CLE(3)
• Active spanning trees with bending energy on planar maps and SLE-decorated Liouville quantum gravity for $\kappa >8$
• Multiple SLE type scaling limits: from local to global
• Connection probabilities for conformal loop ensembles
• UST branches, martingales, and multiple SLE(2)
• On the convergence of FK-Ising Percolation to SLE$(16/3,16/3-6)$
• Connection Probabilities of Multiple FK-Ising Interfaces
• Fermionic observables in the transverse Ising chain
• The fuzzy Potts model in the plane: Scaling limits and arm exponents
• Mating of trees for critical Liouville quantum gravity
• Slit-strip Ising boundary conformal field theory 2: Scaling limits of fusion coefficients
• The bulk one-arm exponent for the CLE${}_{{\kappa }^{\prime }}$ percolations