Conformal Maps and Planar Graphs

共形图和平面图

• 批准号: 1362169
• 负责人: Steffen Rohde
• 金额: $ 18万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-06-15 至 2018-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1362169&HistoricalAwards=false
• 关键词: Conformal; Maps; Planar; Graphs

In many models of statistical physics, such as the Ising model for ferromagnetism, patterns and shapes emerge from randomness. Similar shapes can be observed in various real-life situations, as the models aim to describe natural phenomena. Whereas self-similar sets such as snowflakes look exactly the same at different scales, the random sets the PI studies look only roughly the same at different scales, with their appearance being described by some probabilistic law. While these sets are easy to observe and not hard to simulate, they are difficult to treat mathematically. The last decade has seen dramatic progress in the understanding of such random shapes, but many fundamental questions are still open. The PI's work aims towards resolutions of some of these questions, and tries to provide tools to investigate the shapes of random sets.The aforementioned progress is largely based on conformal invariance properties of the scaling limits, connecting it to the Schramm-Loewner evolution SLE. For instance, by work of Smirnov and others, interfaces of the Ising model at criticality are forms of SLE(3), and the scaling limit of the collection of outer loops is the Conformal Loop Ensemble CLE(3). The PI investigates conformal representations of CLE's using tools from circles packings, aiming at an analog of Bonk's uniformization theorem for(deterministic) Sierpinski carpets. The PI also studies conformal representations of planar graphs using uniformizations of flat surfaces with cone singularities, as well as their representations as Shabat polynomials or Belyi functions. A main goal is to establish the existence of a distributional limit of the conformally natural embedding of random trees as the number of edges tends to infinity. A key tool is the conformal welding of laminations of the disc.

在许多统计物理学模型中，例如铁磁性的ISING模型，模式和形状从随机性中出现。在各种现实生活中，可以观察到类似的形状，因为这些模型旨在描述自然现象。诸如雪花之类的自相似集合在不同的尺度上看起来完全相同，但随机组的PI研究在不同的尺度上看起来大致相同，而某些概率定律则描述了它们的外观。尽管这些集合易于观察，并且不难模拟，但它们很难用数学处理。在过去的十年中，在理解这种随机形状方面取得了巨大的进步，但是许多基本问题仍然开放。 PI的工作旨在解决其中一些问题的分辨率，并试图提供工具来研究随机集的形状。上述进度很大程度上基于缩放限制的保形不变性属性，将其连接到Schramm-Loewner Evolution Evolution SLE。例如，通过Smirnov和其他人的作品，关键性的ISING模型的接口是SLE的形式（3），外圈收集的缩放限制是保形环乐团CLE（3）。 PI使用圆圈包装中的工具调查了CLE的共形表示，旨在以（确定性）Sierpinski地毯的类似物的类似物。 PI还使用具有圆锥形奇异性的平面均匀化及其表示为shabat多项式或贝利函数的平面表面的均匀化研究平面图的形式。一个主要目标是确定随机树的共形自然嵌入的分布极限，因为边缘的数量往往是无穷大的。一个关键工具是椎间盘层压的共形焊接。