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

• 成果类型: Article
• 署名作者: Lawler, GF; Schramm, O; Werner, W
• 署名单位: Cornell University; Universite Paris Saclay; Microsoft
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2004
• 页码: 939-995
• 关键词: self-avoiding walk; brownian intersection exponents; critical percolation; 2 dimensions; VALUES

摘要

This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain D (subset of)(not equal) 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 invariant. Assuming that partial derivativeD 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 subset of partial derivativeD, 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.