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 Probability Surveys > Vol. 8 (2011) open journal systems 


Conformally invariant scaling limits in planar critical percolation

Nike Sun, Stanford University


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 (SLEĸ), 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.

AMS 2000 subject classifications: Primary 60K35, 30C35; secondary 60J65.

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

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Sun, Nike, Conformally invariant scaling limits in planar critical percolation, Probability Surveys, 8, (2011), 155-209 (electronic). DOI: 10.1214/11-PS180.

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