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

我们在这个项目中的目标是研究Ising模型的可积性。伊辛模型是统计力学的一个基本模型，最初用于模拟磁化，但也被用作任何两种共存物质仅局部相互作用的行为的简化模型。它是最简单的数学模型之一，可以显示温度变化时的相变。伊辛模型有着悠久的数学历史，并启发了许多现代数学物理学，如量子场论和弦理论。它的可积性在某种意义上与其大尺度阶性正交。然而，可积性与我们在数学上“解决”模型的能力有关，因此，在这个模型和其他相关模型中，理解可解性和可积性之间的联系是很重要的，这些模型目前似乎很难解决。统计力学是通过概率方法研究由许多相互作用的粒子组成的系统。在制作接近现实世界现象的现实模型和制作通常更抽象的数学上可解决的模型之间存在紧张关系。一个我们希望用来扩大可解模型范围的新工具是可积性的概念。这是一个来自动力系统的概念，在某些情况下，一个复杂的系统可以在适当的坐标变化下表现出非常简单的行为。我们希望在精确意义上理解伊辛模型的可积性，将引导我们在其他经典统计力学模型中看到类似的可能性。

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.