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Conformal Radii for Conformal Loop Ensembles

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Abstract

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.

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Authors and Affiliations

  1. Microsoft Research, One Microsoft Way, Redmond, WA, 98052, USA

    Oded Schramm & David B. Wilson

  2. Courant Institute, New York University, 251 Mercer Street, New York, NY, 10021, USA

    Scott Sheffield

  3. Department of Mathematics, M. I. T., 77 Massachusetts Ave., Cambridge, MA, 02139, USA

    Scott Sheffield

Authors
  1. Oded Schramm
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  2. Scott Sheffield
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  3. David B. Wilson
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Additional information

Communicated by M. Aizenman

Partially supported by NSF grant DMS0403182.

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Schramm, O., Sheffield, S. & Wilson, D.B. Conformal Radii for Conformal Loop Ensembles. Commun. Math. Phys. 288, 43–53 (2009). https://doi.org/10.1007/s00220-009-0731-6

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  • DOI: https://doi.org/10.1007/s00220-009-0731-6

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