Excursion decompositions for SLE and Watts' crossing formula

• 成果类型: Article
• 署名作者: Dubédat, J
• 署名单位: New York University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-005-0446-3
• 发表日期: 2006
• 页码: 453-488
• 关键词: erased random-walks; critical percolation; conformal-invariance

摘要

It is known that Schramm-Loewner Evolutions (SLEs) have a.s. frontier points if kappa > 4 and a.s. cutpoints if 4 < kappa < 8. If kappa > 4, an appropriate version of SLE(kappa) has a renewal property: it starts afresh after visiting its frontier. Thus one call give an excursion decomposition for this particular SLE(kappa) away from its frontier. For 4 < kappa < 8, there is a two-sided analogue of this situation: a particular version of SLE(K) has a renewal property w.r.t its cutpoints; one studies excursion decompositions of this SLE away from its cutpoints. For kappa = 6, this overlaps Virag's results on Brownian beads. As a by-product of this construction. one proves Watts' formula, which describes the probability of a double crossing in a rectangle for critical plane percolation.