RANDOM CURVES, SCALING LIMITS AND LOEWNER EVOLUTIONS

• 成果类型: Article
• 署名作者: Kemppainen, Antti; Smirnov, Stanislav
• 署名单位: University of Helsinki; University of Geneva
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/15-AOP1074
• 发表日期: 2017
• 页码: 698-779
• 关键词: erased random-walks; conformal-invariance; critical percolation; random-cluster; CONVERGENCE; continuity; trees; plane; MODEL; path

摘要

In this paper, we provide a framework of estimates for describing 2D scaling limits by Schramm's SLE curves. In particular, we show that a weak estimate on the probability of an annulus crossing implies that a random curve arising from a statistical mechanics model will have scaling limits and those will be well described by Loewner evolutions with random driving forces. Interestingly, our proofs indicate that existence of a nondegenerate observable with a conformally- invariant scaling limit seems sufficient to deduce the required condition. Our paper serves as an important step in establishing the convergence of Ising and FK Ising interfaces to SLE curves; moreover, the setup is adapted to branching interface trees, conjecturally describing the full interface picture by a collection of branching SLEs.