Communications in Mathematical Physics

, Volume 288, Issue 1, pp 43–53 | Cite as

Conformal Radii for Conformal Loop Ensembles

  • Oded Schramm
  • Scott Sheffield
  • David B. Wilson
Article

Abstract

The conformal loop ensembles CLEκ, defined for 8/3 ≤ κ ≤ 8, are random collections of loops in a planar domain which are conjectured scaling limits of the O(n) loop models. We calculate the distribution of the conformal radii of the nested loops surrounding a deterministic point. Our results agree with predictions made by Cardy and Ziff and by Kenyon and Wilson for the O(n) model. We also compute the expectation dimension of the CLEκgasket, which consists of points not surrounded by any loop, to be
$$2 - \frac{{(8 - \kappa)(3\kappa - 8)}}{{32\kappa}}$$
, which agrees with the fractal dimension given by Duplantier for the O(n) model gasket.

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Oded Schramm
    • 1
  • Scott Sheffield
    • 2
    • 3
  • David B. Wilson
    • 1
  1. 1.Microsoft ResearchOne Microsoft WayRedmondUSA
  2. 2.Courant InstituteNew York UniversityNew YorkUSA
  3. 3.Department of MathematicsM. I. T.CambridgeUSA

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