Translator Disclaimer
January 2004 Conformal invariance of planar loop-erased random walks and uniform spanning trees
Gregory F. Lawler, Oded Schramm, Wendelin Werner
Ann. Probab. 32(1B): 939-995 (January 2004). DOI: 10.1214/aop/1079021469

Abstract

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 SLE$_2$ 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 $\p D$ is a $C^1$-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 $A\subset\p D$, is the chordal SLE$_8$ path in $\overline D$ joining the endpoints of A. A by-product of this result is that SLE$_8$ is almost surely generated by a continuous path. The results and proofs are not restricted to a particular choice of lattice.

Citation

Download Citation

Gregory F. Lawler. Oded Schramm. Wendelin Werner. "Conformal invariance of planar loop-erased random walks and uniform spanning trees." Ann. Probab. 32 (1B) 939 - 995, January 2004. https://doi.org/10.1214/aop/1079021469

Information

Published: January 2004
First available in Project Euclid: 11 March 2004

zbMATH: 1126.82011
MathSciNet: MR2044671
Digital Object Identifier: 10.1214/aop/1079021469

Subjects:
Primary: 82B41

Rights: Copyright © 2004 Institute of Mathematical Statistics

JOURNAL ARTICLE
57 PAGES


SHARE
Vol.32 • No. 1B • January 2004
Back to Top