Critical planar systems and conformal invariance

临界平面系统和共形不变性

• 批准号: 1005749
• 负责人: Julien Dubedat
• 金额: $ 16.51万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1005749&HistoricalAwards=false
• 关键词: Critical; planar; systems; conformal; invariance

The last decade has seen a series of major achievements in the study of conformally invariant systems, in particular following the introduction of Schramm-Loewner Evolutions. This new approach has proved extremely effective in describing the scaling limit of interfaces of critical models of two-dimensional Statistical Physics, such as percolation and the Ising model, in accordance with the earlier predictions of Conformal Field Theory (CFT). The research program proposed here focuses on systems on paths and loops, in particular in relation with the CFT formalism; on free field invariance principles and the analysis of Cauchy-Riemann operators; and on the connections between geometric and functional representations of continuous scaling limits.The goal of this proposal is to analyze mathematical models of the physical phenomenon of phase transition. A phase transition describes a sharp qualitative change in a physical system under variation of an external parameter, such as freezing of water (transition from liquid to solid phase as temperature decreases). At the phase transition, a random macroscopic geometry may emerge, akin to the wiggly interfaces separating non mixing fluids like oil and water, or positively and negatively charged regions in a magnetic material. The study of those fluctuating interfaces is grounded in both Probability Theory and Theoretical Physics. Special emphasis will be put on the interplay between these two approaches, and their relation to other areas of mathematics.

在过去的十年里，共形不变系统的研究取得了一系列重大成就，特别是在Schramm-Loewner演化之后。这种新的方法已被证明是非常有效的描述界面的临界模型的二维统计物理，如渗流和伊辛模型的标度极限，根据共形场论（CFT）的早期预测。这里提出的研究计划侧重于系统的路径和回路，特别是与CFT形式主义;自由场不变性原则和柯西-黎曼算子的分析;和连续标度极限的几何和功能表示之间的连接。这个建议的目标是分析相变的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变，例如水的冻结（随着温度降低从液相到固相的转变）。在相变时，可能会出现随机的宏观几何形状，类似于分离非混合流体（如油和水）或磁性材料中的正电荷和负电荷区域的摇摆界面。这些起伏界面的研究是建立在概率论和理论物理学的基础上的。将特别强调这两种方法之间的相互作用，以及它们与其他数学领域的关系。