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

Wendelin Werner, Oded Schramm, Gregory F. Lawler
2004 Annals of Probability  
This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain D ~ tC 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 aD is a C 1 -simp1e 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
more » ... ree is wired along a proper arc A c aD, is the chordal SLEg path in I5 joining the endpoints of A. A by-product of this result is that SLEg is almost surely generated by a continuous path. The results and proofs are not restricted to a particular choice of lattice.
doi:10.1214/aop/1079021469 fatcat:lmhxn7hjxbfpnesgewaudm3fu4