The Full Scaling Limit of Two-Dimensional Critical Percolation

TL;DR

This paper establishes the full scaling limit of two-dimensional critical percolation as a continuum loop process, confirming conformal invariance and connecting discrete models to SLE(6) paths.

Contribution

It constructs the continuum loop process from SLE(6) and proves it matches the scaling limit of 2D critical percolation interfaces, including a detailed convergence proof.

Findings

The continuum loop process is conformally invariant.

The scaling limit of percolation interfaces is described by SLE(6).

A rigorous proof of convergence to SLE(6) is provided.

Abstract

We use SLE(6) paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice -- that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved. In the main body of the paper these results are proved while assuming, as argued by Schramm and Smirnov, that the percolation exploration path converges in distribution to the trace of chordal SLE(6). Then, in a lengthy appendix, a detailed proof is provided for this convergence to SLE(6), which itself relies on Smirnov's result that crossing probabilities converge to Cardy's formula.