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Critical percolation exploration path and SLE6: a proof of convergence
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  • Published: 21 March 2007

Critical percolation exploration path and SLE6: a proof of convergence

  • Federico Camia1 &
  • Charles M. Newman2 

Probability Theory and Related Fields volume 139, pages 473–519 (2007)Cite this article

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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 SLE 6. 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 SLE 6 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.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands

    Federico Camia

  2. Courant Institute of Mathematical Sciences, New York University, New York, USA

    Charles M. Newman

Authors
  1. Federico Camia
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  2. Charles M. Newman
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Corresponding author

Correspondence to Federico Camia.

Additional information

Research of Federico Camia was partially supported by a Marie Curie Intra-European Fellowship under contract MEIF-CT-2003-500740 and by a Veni grant of the Dutch Organization for Scientific Research (NWO).

Research of Charles M.Newman was partially supported by the US NSF under grants DMS-01-04278 and DMS-06-06696.

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Camia, F., Newman, C.M. Critical percolation exploration path and SLE6: a proof of convergence. Probab. Theory Relat. Fields 139, 473–519 (2007). https://doi.org/10.1007/s00440-006-0049-7

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  • Received: 25 April 2006

  • Revised: 22 November 2006

  • Published: 21 March 2007

  • Issue Date: November 2007

  • DOI: https://doi.org/10.1007/s00440-006-0049-7

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Keywords

  • Continuum scaling limit
  • Percolation
  • SLE
  • Critical behavior
  • Triangular lattice
  • Conformal invariance

Mathematics Subject Classification (2000)

  • 82B27
  • 60K35
  • 82B43
  • 60D05
  • 30C35
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