Probability Surveys

Conformally invariant scaling limits in planar critical percolation

Nike Sun

Full-text: Open access

Abstract

This is an introductory account of the emergence of conformal invariance in the scaling limit of planar critical percolation. We give an exposition of Smirnov’s theorem (2001) on the conformal invariance of crossing probabilities in site percolation on the triangular lattice. We also give an introductory account of Schramm-Loewner evolutions (SLEκ), a one-parameter family of conformally invariant random curves discovered by Schramm (2000). The article is organized around the aim of proving the result, due to Smirnov (2001) and to Camia and Newman (2007), that the percolation exploration path converges in the scaling limit to chordal SLE6. No prior knowledge is assumed beyond some general complex analysis and probability theory.

Article information

Source
Probab. Surveys, Volume 8 (2011), 155-209.

Dates
First available in Project Euclid: 28 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.ps/1319806861

Digital Object Identifier
doi:10.1214/11-PS180

Mathematical Reviews number (MathSciNet)
MR2846901

Zentralblatt MATH identifier
1245.60096

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 30C35: General theory of conformal mappings
Secondary: 60J65: Brownian motion [See also 58J65]

Keywords
Conformally invariant scaling limits percolation Schramm-Loewner evolutions preharmonicity preholomorphicity percolation exploration path

Citation

Sun, Nike. Conformally invariant scaling limits in planar critical percolation. Probab. Surveys 8 (2011), 155--209. doi:10.1214/11-PS180. https://projecteuclid.org/euclid.ps/1319806861


Export citation

References