Conformally invariant scaling limits in planar critical percolation

Details Conformally invariant scaling limits in planar critical percolation Conformally invariant scaling limits in planar critical percolation

2009-11-02

arxiv.org/abs/0911.0063v2

Probability Surveys, Vol. 8 (2011) 155-209

Mathematics

Probability

55 pages, 10 figures

Scientific paper

10.1214//11-PS180

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 (SLE(k)), 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 SLE(6). No prior knowledge is assumed beyond some general complex analysis and probability theory.

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