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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

The conformally invariant measure on self-avoiding loops


Author: Wendelin Werner
Journal: J. Amer. Math. Soc. 21 (2008), 137-169
MSC (2000): Primary 60D05; Secondary 82B41, 82B43, 30C99, 60J65
Published electronically: February 20, 2007
MathSciNet review: 2350053
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Abstract: We show that there exists a unique (up to multiplication by constants) and natural measure on simple loops in the plane and on each Riemann surface, such that the measure is conformally invariant and also invariant under restriction (i.e. the measure on a Riemann surface $ S'$ that is contained in another Riemann surface $ S$ is just the measure on $ S$ restricted to those loops that stay in $ S'$). We study some of its properties and consequences concerning outer boundaries of critical percolation clusters and Brownian loops.


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Additional Information

Wendelin Werner
Affiliation: Université Paris-Sud, Laboratoire de Mathématiques, Université Paris-Sud, Bât. 425, 91405 Orsay cedex, France and DMA, Ecole Normale Supérieure, 45 rue d’Ulm, 75230 Paris cedex, France
Email: wendelin.werner@math.u-psud.fr

DOI: http://dx.doi.org/10.1090/S0894-0347-07-00557-7
PII: S 0894-0347(07)00557-7
Received by editor(s): December 17, 2005
Published electronically: February 20, 2007
Additional Notes: This work was supported by the Institut Universitaire de France
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.