Integrability and limit shapes in the two-dimensional Ising model and related models

二维伊辛模型及相关模型中的可积性和极限形状

• 批准号: 1208191
• 负责人: Richard Kenyon
• 金额: $ 38.99万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1208191&HistoricalAwards=false
• 关键词: Integrability; limit; shapes; two; dimensional

Our goal in this project is to study the integrability properties of the Ising model. The Ising model is a fundamental model of statistical mechanics, originally intended to model magnetization, but is also used as a simplified model for the behavior of any two co-existing substances interacting only locally. It is one of the simplest mathematical models which displays a phase transition as the temperature is varied. The Ising model has a long mathematical history and has inspired much of modern mathematical physics such as quantum field theory and string theory. Its integrability properties act in some sense orthogonally to its large-scale order properties. The integrability is nonetheless related to our ability to "solve" the model mathematically, so the connection between solvability and integrability is important to understand in this and other related models, which seem at present much harder to solve. Statistical mechanics is concerned with studying, through probabilistic methods, systems of many interacting particles. There is a tension between making realistic models, which are close to real-world phenomena, and making mathematically solvable models, which are typically more abstract. A new tool which we hope to use to increase the range of our solvable models is the notion of integrability. This is the notion, coming from dynamical systems, that in certain cases a complicated system can be shown to exhibit in fact quite simple behavior under an appropriate change of coordinates. We hope that understanding in a precise sense the integrability properties of the Ising model will lead us to see similar possibilities in other classical statistical mechanical models.

我们在这个项目中的目标是研究Ising模型的可合转属性。 ISING模型是统计力学的基本模型，最初旨在模拟磁化，但也被用作简化的模型，用于仅在本地相互作用的任何两个共存物质的行为。它是最简单的数学模型之一，随着温度的变化而显示相变。 Ising模型具有悠久的数学历史，并激发了许多现代数学物理学，例如量子场理论和弦理论。它的集成性属性在某种意义上与其大规模订单属性正交起来。尽管如此，该集成性仍然与我们通过数学“求解”模型的能力有关，因此在该和其他相关模型中，可溶性和集成性之间的联系很重要，目前似乎更难解决。统计力学与通过概率方法和许多相互作用颗粒的系统进行研究有关。制作现实模型（与现实世界现象接近的现实模型与制造数学上可解决的模型之间）之间存在张力，通常是更抽象的。我们希望用来增加可解决模型范围的新工具是集成性的概念。这是来自动态系统的概念，在某些情况下，可以证明复杂的系统在适当的坐标变化下表现出非常简单的行为。我们希望从精确的意义上理解ISING模型的整合性能将使我们在其他经典的统计机械模型中看到类似的可能性。