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

2006/11/05 by Federico Camia, Charles M. Newman, Camia, Federico +1 · 2 citations
Mathematics · Physics and Astronomy · #30C35 #60D05 #60K35 #82B27 #82B43 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #NMR spectroscopy and applications #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.stat-mech #math-ph #math.MP #math.PR #msc:30C35 #msc:60D05 #msc:60K35 #msc:82B27 #msc:82B43

paper · pdf · doi:10.48550/arxiv.math/0611116

23 pages, 6 figures

arxiv created 2006/11/05 · openalex publication_date 2006/11/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Two-Dimensional Scaling Limits via Marked Nonsimple Loops

Cited by

• Liouville quantum gravity: from random planar maps to conformal field theory
• Random walk reflected off of infinity, with applications to uniform spanning forests and supercritical Liouville quantum gravity

Related

• The Full Scaling Limit of Two-Dimensional Critical Percolation
• Scaling Limits of Two-Dimensional Percolation: an Overview
• Critical Percolation Exploration Path and SLE(6): a Proof of Convergence
• Two-Dimensional Scaling Limits via Marked Nonsimple Loops
• Critical percolation: the expected number of clusters in a rectangle