Critical and near critical systems in statistical mechanics

这个项目将从统计力学和概率论的角度研究表现出关键现象的系统。这些系统包括自避免随机漫步和循环、伊辛旋转系统和渗透。我们的第一种方法是通过实空间重整化群变换。这些Ising类型模型的转换将模型的关键行为与临界点附近的重整化群映射的行为联系起来。这张图在数学上没有很好的定义，该项目将研究这张图的数学性质。我们的第二种方法仅限于二维系统，当存在临界行为时，保角不变性奇迹般地出现。本研究将通过研究Loewner驱动过程来研究Schramm-Loewner演化与各种离散模型之间的关系，特别是对于接近而非处于临界点的模型。研究了双无限自避行走与新发现的平面上自避环的保形不变测度之间的关系。将要研究的系统包含微观长度尺度上的随机性。对于大多数参数值，例如温度，在宏观尺度上看不到这种随机性。但对于参数的特殊值，这种微观随机性可以产生宏观效应。这是相变的一个特征。物理学家已经开发出了研究相变的强大技术，但这些方法的数学原理还没有得到很好的理解。这项研究将通过发展物理学家最重要的两个工具——重整化群和共形不变性——背后的数学，进一步加深我们对关键现象的理解。重整化组的研究将集中在Ising模型上，这可以说是晶体材料磁性行为的最重要的单一模型。共形不变性研究的一个成果将是一种更快的共形映射数值计算算法，这种算法在科学和工程中广泛使用。

This project will study systems from statistical mechanics and probability that exhibit critical phenomena. The systems include self-avoiding random walks and loops, Ising spin systems, and percolation. Our first approach is through real space renormalization group transformations. These transformations for Ising type models relate the critical behavior of the model to the behavior of the renormalization group map near the critical point. This map is not well defined mathematically, and the project will study the mathematical properties of this map. Our second approach is restricted to two dimensional systems where conformal invariance miraculously appears when there is critical behavior. The research will study the relation between the Schramm-Loewner evolution and various discrete models by studying the Loewner driving process, especially for models which are near but not at their critical point. The relation of the bi-infinite self-avoiding walk to the newly discovered conformally invariant measures on self-avoiding loops in the plane will also be investigated.The systems that will be studied contain randomness at a microscopic length scale. For most values of the parameters, e.g., temperature, this randomness is not seen at the macroscopic scale. But for special values of the parameters this microscopic randomness can produce macroscopic effects. This is one characterization of a phase transition. Physicists have developed powerful techniques for studying phase transitions, but the mathematics of these methods is not well understood. This research will further our understanding of critical phenomena by developing the mathematics behind two of the most important of the physicist's tools - the renormalization group and conformal invariance. The research on the renormalization group will focus on the Ising model, arguably the single most important model of the magnetic behavior of crystalline materials. One product of the research on conformal invariance will be a much faster algorithm for numerically computing conformal maps which are used extensively in science and engineering.