Probability Theory and Related Fields

, Volume 139, Issue 3, pp 473–519

Critical percolation exploration path and SLE6: a proof of convergence

Open AccessArticle

DOI: 10.1007/s00440-006-0049-7

Cite this article as:
Camia, F. & Newman, C.M. Probab. Theory Relat. Fields (2007) 139: 473. doi:10.1007/s00440-006-0049-7

Abstract

It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE6. We provide here a detailed proof, which relies on Smirnov’s theorem that crossing probabilities have a conformally invariant scaling limit (given by Cardy’s formula). The version of convergence to SLE6 that we prove suffices for the Smirnov–Werner derivation of certain critical percolation crossing exponents and for our analysis of the critical percolation full scaling limit as a process of continuum nonsimple loops.

Keywords

Continuum scaling limitPercolationSLECritical behaviorTriangular latticeConformal invariance

Mathematics Subject Classification (2000)

82B2760K3582B4360D0530C35
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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsVrije Universiteit AmsterdamAmsterdamThe Netherlands
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA