Conformal Measure Ensembles for Percolation and the FK-Ising model

TL;DR

This paper constructs and analyzes the scaling limits of open clusters and their measures in 2D percolation models, including FK-Ising, revealing conformal covariance and geometric properties of the continuum limits.

Contribution

It introduces a unified framework for the scaling limits of clusters and measures in 2D percolation and FK-Ising models, establishing conformal covariance and continuum representations.

Findings

Scaling limit of open clusters constructed and characterized.

Conformal covariance of the scaling limit established.

Existence and uniqueness of the continuum magnetization field proven.

Abstract

Under some general assumptions, we construct the scaling limit of open clusters and their associated counting measures in a class of two dimensional percolation models. Our results apply, in particular, to critical Bernoulli site percolation on the triangular lattice and to the critical FK-Ising model on the square lattice. Fundamental properties of the scaling limit, such as conformal covariance, are explored. As an application to Bernoulli percolation, we obtain the scaling limit of the largest cluster in a bounded domain. We also apply our results to the critical, two-dimensional Ising model, obtaining the existence and uniqueness of the scaling limit of the magnetization field, as well as a geometric representation for the continuum magnetization field which can be seen as a continuum analog of the FK representation.