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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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The conformally invariant measure on self-avoiding loops
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by Wendelin Werner
J. Amer. Math. Soc. 21 (2008), 137-169
Published electronically: February 20, 2007


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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Bibliographic 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:
  • Received by editor(s): December 17, 2005
  • Published electronically: February 20, 2007
  • Additional Notes: This work was supported by the Institut Universitaire de France
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 21 (2008), 137-169
  • MSC (2000): Primary 60D05; Secondary 82B41, 82B43, 30C99, 60J65
  • DOI:
  • MathSciNet review: 2350053