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


The conformal loop ensembles CLE κ , 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 CLE κ gasket, 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.


Moment Generate Function Loop Model Local Martingale Critical Percolation Conformal Radius 
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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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