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 \(D\) 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.