Commutation Relations for SLE in Multiply Connected Domains and the Coupling Technique

该奖项是根据2009年美国复苏和再投资法案(公法111-5)资助的。该项目旨在研究由Oded Schramm引入的Schramm-Loewner进化(SLE)的几何性质。SLE描述了平面区域中的共形不变随机分形曲线，这些曲线是许多有趣的二维统计晶格模型在临界温度下的标度极限，如渗流、伊辛模型、高斯自由场。SLE与共形场论之间的关系已经建立并得到了很好的理解。对SLE本身的研究有助于人们更好地理解这些格子模型和共形场论。该项目广泛地使用了所谓的耦合技术，成功地证明了弦式SLE的可逆性猜想和关于SLE边界的Duplan tier-S对偶猜想。这项技术使人们能够构建两条SLE曲线的全局耦合，如果它们彼此通勤的话。PI计划使用耦合技术来研究定义在多连通域中的SLE，这一点很重要，因为许多晶格模型都是在那里自然定义的。这些格子模型的性质表明，任何合理定义的SLE都必须满足对易关系，因此耦合技术可以用来构造全局对易耦合。该项目的重点是定义在具有几个标记边界点的双连通域中的SLE。这些SLE与这些领域中的各种晶格模型有着天然的联系，它们可以用来证明径向SLE的可逆性。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project aims at studying the geometric properties of Schramm-Loewner evolution (SLE) introduced by Oded Schramm. SLE describes conformal invariant random fractal curves in plane domains, which are the scaling limits of many interesting two dimensional statistical lattice models at critical temperature, e.g., percolation, Ising model, Gaussian free field. The relationship between SLE and Conformal Field Theory (CFT) has been established and well understood.The study of SLE itself helps people to gain better understanding of those lattice models and CFT.The project extensively uses the so-called coupling technique, which has been successful in proving the reversibility conjecture for chordal SLE and Duplantier?s duality conjecture about the boundary of SLE. The technique enables one to construct a global coupling of two SLE curves if they commute with each other. The PI plans to use the coupling technique to study SLE defined in multiply connected domains, which are important because many lattice models are naturally defined there. The properties of these lattice models indicate that any reasonablely defined SLE must satisfy commutation relation, and so the coupling technique could be applied to construct a global commutation coupling. The project focuses on SLE defined in doubly connected domains with a few marked boundary points. These SLE naturally relate with various lattice models in these domains, and they could be used to prove the reversibility of radial SLE.