Conformally invariant random systems

近年来，人们对二维统计物理学的理解取得了重要进展。schramm - loewner进化（SLE）及其相关对象的引入极大地促进了对微观随机输入和相互作用产生的宏观随机几何结构的严谨研究。这使得在理论物理学中建立共形场论（CFT）预测的结果成为可能，大多数情况下没有与技术相匹配。侦探的研究计划主要是针对实现基于SLE结构的CFT概念。这些方法和问题涉及几何、泛函分析和表示理论方面的相互作用。本提案的目的是分析相变物理现象的数学模型。相变描述了物理系统在外部参数变化下的急剧质的变化，例如水的冻结（随着温度的降低从液态转变为固态）。在相变时，可能会出现随机的宏观几何形状，类似于将油和水等非混合流体分开的弯曲界面。对这些波动界面的研究植根于概率论和理论物理学。我们将特别关注这两种方法之间的相互作用，以及分析中涉及的其他数学领域。

Recent years have seen important progress in the understanding of twodimensional statistical physics. The rigorous study of macroscopicrandom geometric structures generated by microscopic random inputs andinteractions has been greatly stimulated by the introduction ofSchramm-Loewner Evolutions (SLE) and related objects. This enabled toestablish results predicted by Conformal Field Theory (CFT) in TheoreticalPhysics, most often without matching the techniques. The P.I.'s research programis primarily directed at realizing CFT concepts from SLE basedconstructions. The methods and issues involve an interplay ofgeometric, functional analytic and representation theoretic aspects.The goal of this proposal is to analyze mathematical models of thephysical phenomenon of phase transition. A phase transition describes a sharpqualitative change in a physical system under variation of an externalparamater, such as freezing of water (transition from liquid tosolid phase as temperature decreases). At the phase transition, a random macroscopic geometrymay emerge, akin to the wiggly interfaces separating non mixing fluidslike oil and water. The study of those fluctuating interfaces isrooted in both Probability Theory and Theoretical Physics. Special focuswill be placed on the interaction between these two approaches, andother areas of mathematics involved in the analysis.