The conformally invariant measure on self-avoiding loops
- 20 February 2007
- journal article
- Published by American Mathematical Society (AMS) in Journal of the American Mathematical Society
- Vol. 21 (1) , 137-169
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 ′ S’ that is contained in another Riemann surface S S is just the measure on S S restricted to those loops that stay in S ′ S’ ). We study some of its properties and consequences concerning outer boundaries of critical percolation clusters and Brownian loops.Keywords
All Related Versions
This publication has 35 references indexed in Scilit:
- Basic properties of SLEAnnals of Mathematics, 2005
- Conformal restriction and related questionsProbability Surveys, 2005
- Girsanov’s transformation for SLE processes, intersection exponents and hiding exponentsAnnales de la Faculté des sciences de Toulouse : Mathématiques, 2004
- Conformal invariance of planar loop-erased random walks and uniform spanning treesThe Annals of Probability, 2004
- SLEs as boundaries of clusters of Brownian loopsComptes Rendus Mathematique, 2003
- SLE martingales and the Virasoro algebraPhysics Letters B, 2003
- Path-Crossing Exponents and the External Perimeter in 2D PercolationPhysical Review Letters, 1999
- Sur la forme des composantes connexes du complémentaire de la courbe brownienne planeProbability Theory and Related Fields, 1994
- Nonintersection Exponents for Brownian Paths. II. Estimates and Applications to a Random FractalThe Annals of Probability, 1990
- Cut Points on Brownian PathsThe Annals of Probability, 1989