Conformal Invariance Of Planar Loop-Erased Random Walks and Uniform Spanning Trees

Abstract

This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain D ~ tC 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 aD is a C 1 -simp1e 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 c aD, is the chordal SLEg path in I5 joining the endpoints of A. A by-product of this result is that SLEg is almost surely generated by a continuous path. The results and proofs are not restricted to a particular choice of lattice.

Imaginary geometry I: interacting SLEs

“…The boundary data for h is chosen so that the central ("north-going") curve shown should approximate an SLE 1/8 process (color figure online) . The boundary data for h is chosen so that the central ("north-going") curve shown should approximate an SLE 1 process (color figure online) duality [i.e., descriptions of the boundaries of SLE 16/κ (ρ) curves], along with a "light cone" interpretation of SLE 16/κ (ρ) that allows these curves to be constructed and decomposed in surprising ways.…”

“…This approach (and a range of θ values) was used to generate the rays in Figs. 2,3,4,5,7,8,13,14,15,16,17,18, and 21. We expect that these rays will converge to limiting path-valued functions of h as the mesh size gets finer.…”

“…We accomplish this by taking η to be coupled with h and η to be coupled with −h, as in the statement of Theorem 1.1 (this is the reason we can always take χ > 0) (Figs. 13,14,15,16,17,18,19).…”

“…SLE κ is a one-parameter family of conformally invariant random curves, introduced by Schramm [27] as a candidate for (and later proved to be) the scaling limit of loop erased random walk [16] and the interfaces in critical percolation [2,33]. Schramm's curves have been shown so far also to arise as the scaling limit of the macroscopic interfaces in several other models from statistical physics: [3,17,[34][35][36].…”

Markov Paths, Loops and Fields

“…The extension of the properties proved here in the finite framework has still to be completed, though the relation with spanning trees should follow from the remarkable results obtained on SLE processes, especially [22]. Note finally that other essential relations between SLE processes, loops and free fields appear in [59], [42] [7], and more recently in [45] and [46].…”