超立方晶格上的Ising和Potts模型讲座
Lectures on the Ising and Potts models on the hypercubic lattice
摘要：
相变是统计力学的一个中心主题，更广泛地说，是概率论的一个中心主题。晶格自旋模型代表了有限维相变的一般范式，描述了铁磁体甚至一些流体（晶格气体）。自20世纪80年代以来，人们已经认识到，随机几何表示，如随机游动和随机电流表示，是理解自旋模型的有力工具。除了自旋模型固有的技术之外，这种表示还提供了从渗流理论获得丰富思想的途径。近年来，对于二维自旋模型，这些思想已经进一步与离散复分析的思想相结合。通过这些连接得到的壮观的结果包括证明，界面的二维伊辛模型具有共形不变的尺度限制SLE曲线，该连接常数的自避免行走的六边形晶格是由$\sqrt{2+\sqrt 2}$。在更高的维度，理解也进展的证明，波茨模型的相变是尖锐的，三维伊辛模型的磁化消失在临界点。
英文摘要：
Phase transitions are a central theme of statistical mechanics, and of probability more generally. Lattice spin models represent a general paradigm for phase transitions in finite dimensions, describing ferromagnets and even some fluids (lattice gases). It has been understood since the 1980s that random geometric representations, such as the random walk and random current representations, are powerful tools to understand spin models. In addition to techniques intrinsic to spin models, such representations provide access to rich ideas from percolation theory. In recent years, for two-dimensional spin models, these ideas have been further combined with ideas from discrete complex analysis. Spectacular results obtained through these connections include the proofs that interfaces of the two-dimensional Ising model have conformally invariant scaling limits given by SLE curves, that the connective constant of the self-avoiding walk on the hexagonal lattice is given by $\sqrt{2+\sqrt 2}$. In higher dimensions, the understanding also progresses with the proof that the phase transition of Potts models is sharp, and that the magnetization of the three-dimensional Ising model vanishes at the critical point.
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