Conformal invariance of planar loop-erased random walks and uniform spanning trees

• Gregory F. Lawler ^[1] ; Oded Schramm ^[3] ; Wendelin Werner ^[2]
  1. [1] Cornell University

     Cornell University

     City of Ithaca, Estados Unidos

     
  2. [2] Université Paris - Sud
  3. [3] Microsoft Corporation

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 32, Nº. 1, 2, 2004, págs. 939-995
• Idioma: inglés
• DOI: 10.1214/aop/1079021469
• Texto completo no disponible (Saber más ...)
• Resumen

  • This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain Dsubsetneqq\C is equal to the radial SLE2 path. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invariant. Assuming that \pD is a C1-simple closed curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano curve, where the tree is wired along a proper arc A⊂\pD, is the chordal SLE8 path in D¯¯¯¯ joining the endpoints of A. A by-product of this result is that SLE8 is almost surely generated by a continuous path. The results and proofs are not restricted to a particular choice of lattice.