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 \(D\mathop \subset \limits_ \ne \mathbb{C} \) 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 invarint. Assuming that ∂D 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 are A ⊂ ∂ D, is the chordal SLE8 path in \(\overline D \) joining the endpoints of A. A by-product of this result is that SLE8 is almost surely generated by a continuous path. The result and proofs are not restricted to particular choice of Iattice.

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