The Annals of Probability

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

Gregory F. Lawler, Oded Schramm, and Wendelin Werner

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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.

Article information

Ann. Probab. Volume 32, Number 1B (2004), 939-995.

First available in Project Euclid: 11 March 2004

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Zentralblatt MATH identifier

Primary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Loop-erased random walk uniform spanning trees stochastic Loewner evolution


Lawler, Gregory F.; Schramm, Oded; Werner, Wendelin. Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 (2004), no. 1B, 939--995. doi:10.1214/aop/1079021469.

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