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

Abstract

We prove that the scaling limit of loop-erased random walk in a simply connected domain $D$ is equal to the radial SLE(2) path in $D$. 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 the boundary of the domain 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 $A$ on the boundary, is the chordal SLE(8) path in the closure of $D$ 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.