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

We prove that the scaling limit of 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 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. Duke University and Cornell University; partially supported by the National Science Foundation and the Mittag-Leffler Institute. Microsoft Research. Université Paris-Sud and IUF.