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伊辛模型的可积性。伊辛模型是统计力学的基本模型，最初的目的是对磁化进行建模，但也被用作任何两个共存物质仅在局部相互作用的行为的简化模型。这是最简单的数学模型之一，它显示了随着温度的变化而发生的相变。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，如量子场论和弦理论。它的可积性性质在某种意义上与它的大尺度序性质是垂直的。尽管如此，可积性与我们用数学方法“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，这些模型目前看起来更难解决。统计力学致力于通过概率方法研究许多相互作用的粒子系统。在制作接近真实世界现象的逼真模型和制作通常更抽象的数学可解模型之间存在着紧张。我们希望用来扩大可解模型范围的一个新工具是可积性的概念。这是一种来自动力系统的概念，即在某些情况下，一个复杂的系统可以在适当的坐标变化下表现出实际上非常简单的行为。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.