SLE(6) and CLE(6) from Critical Percolation

Abstract

We review some of the recent progress on the scaling limit of two-dimensional critical percolation; in particular, the convergence of the exploration path to chordal SLE(6) and the "full" scaling limit of cluster interface loops. The results given here on the full scaling limit and its conformal invariance extend those presented previously. For site percolation on the triangular lattice, the results are fully rigorous. We explain some of the main ideas, skipping most technical details.

23 pages, 6 figures

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• Conformal Invariance and Percolation
• The Scaling Limit Geometry of Near-Critical 2D Percolation
• Some recent aspects of random conformally invariant systems
• On the Red-Green-Blue Model
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Cited by in corpus (11)

• Conformal Correlation Functions in the Brownian Loop Soup
• Ising (Conformal) Fields and Cluster Area Measures
• Integrability of Conformal Loop Ensemble: Imaginary DOZZ Formula and Beyond
• Conformal loop ensembles and the stress-energy tensor. I. Fundamental notions of CLE
• On Convergence to SLE${}_6$ I: Conformal Invariance for Certain Models of the Bond-Triangular Type
• On Convergence to SLE${}_6$ II: Discrete Approximations and Extraction of Cardy's Formula for General Domains
• Brownian Loops and Conformal Fields
• Limit theorems for critical first-passage percolation on the triangular lattice
• Mating of trees for critical Liouville quantum gravity
• Towards Conformal Invariance and a Geometric Representation of the 2D Ising Magnetization Field
• Multi-arm incipient infinite clusters in 2D: scaling limits and winding numbers