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Conformally invariant scaling limits in planar critical percolation. (English) Zbl 1245.60096
Summary: 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 [S. Smirnov, C. R. Acad. Sci., Paris, Sér. I, Math. 333, No. 3, 239–244 (2001; Zbl 0985.60090)] on the conformal invariance of crossing probabilities in site percolation on the triangular lattice. We also give an introductory account of Schramm-Loewner evolutions \((\text{SLE}_{\kappa })\), a one-parameter family of conformally invariant random curves discovered by O. Schramm [Isr. J. Math. 118, 221–288 (2000; Zbl 0968.60093)]. The article is organized around the aim of proving the result, due to Smirnov [loc. cit.] and to F. Camia and C. M. Newman [Probab. Theory Relat. Fields 139, No. 3–4, 473–519 (2007; Zbl 1126.82007)], that the percolation exploration path converges in the scaling limit to chordal \(\text{SLE}_{6}\). No prior knowledge is assumed beyond some general complex analysis and probability theory.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
30C35 General theory of conformal mappings
60J65 Brownian motion
82B27 Critical phenomena in equilibrium statistical mechanics
82B43 Percolation
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