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

Abstract

This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain D ⊂≠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 ∂ D 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 ⊂ ∂ D, is the chordal SLE8 path in D̄ 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.