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

This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain Dsubsetneqq 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.