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，例如铁磁性的伊辛模型，图案和形状都是随机产生的。类似的形状可以在各种现实生活中观察到，因为模型旨在描述自然现象。自相似集合（如雪花）在不同尺度下看起来完全相同，而PI研究的随机集合在不同尺度下看起来只是大致相同，它们的外观由某种概率定律描述。虽然这些集合很容易观察，也不难模拟，但它们很难用数学方法处理。在过去的十年里，人们对这种随机形状的理解取得了巨大的进步，但许多基本问题仍然悬而未决。PI的工作旨在解决其中的一些问题，并试图提供工具来研究随机集的形状。上述进展主要基于标度极限的共形不变性，将其与Schramm-Loewner演化SLE联系起来。例如，通过斯米尔诺夫等人的工作，伊辛模型在临界状态下的界面是SLE（3）的形式，而外部回路集合的标度极限是共形回路包络（3）。PI调查共形表示CLE的使用工具，从圆包装，旨在模拟邦克的均匀化定理（确定性）谢尔宾斯基地毯。PI还研究了平面图的保角表示，使用具有锥奇点的平坦表面的均匀化，以及它们作为Shabat多项式或Belyi函数的表示。一个主要的目标是建立一个分布极限的共形自然嵌入随机树的边缘数趋于无穷大。一个关键的工具是盘的叠片的保形焊接。