Conformal invariance in random cluster models. II. Full scaling limit as a branching SLE

TL;DR

This paper proves that the interface loops in the random cluster Ising model converge to a conformal loop ensemble, with the exploration tree converging to a branching SLE, providing a detailed geometric description of the model's scaling limit.

Contribution

It establishes the convergence of the loop ensemble to CLE and the exploration tree to a branching SLE, advancing understanding of the model's conformal invariance and geometric structure.

Findings

Loop ensemble converges to conformal loop ensemble (CLE).

Exploration tree converges to branching SLE(16/3, -2/3).

Branching SLE enjoys locality and arises from Ising observables.

Abstract

In the second article of this series, we establish the convergence of the loop ensemble of interfaces in the random cluster Ising model to a conformal loop ensemble (CLE) --- thus completely describing the scaling limit of the model in terms of the random geometry of interfaces. The central tool of the present article is the convergence of an exploration tree of the discrete loop ensemble to a branching SLE$(16/3,-2/3)$. Such branching version of the Schramm's SLE not only enjoys the locality property, but also arises logically from the Ising model observables.