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

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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.