Limit Shapes from a Combinatorial Viewpoint

从组合角度限制形状

基本信息

批准号：
1939926
负责人：
Richard Kenyon
金额：
$ 9.04万
依托单位：
Yale University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1939926&HistoricalAwards=false
关键词：
Limit Shapes Combinatorial Viewpoint

项目摘要

How does water turn into ice? How does a collection of iron atoms gain or lose magnetic properties with changes in temperature? "Statistical mechanics" is the branch of mathematics and physics which deals with understanding such systems: systems consisting of large numbers of identical objects, interacting locally. The main goals of statistical mechanics are to describe large-scale behavior and phase transitions through mathematical methods. One of the most interesting and important types of behavior is when an external force, such as an imposed boundary condition or other constraint, results in a non-homogeneity in the resulting ensemble. A simple example of this is the water/ice transition under a pressure gradient induced by gravity. The principal investigator proposes to study mathematical models of such behavior in a variety of basic settings, in hopes of understanding their common features and to be able to predict similar behavior in other potentially more complex systems.The notion of limit shape in probability describes the property of large random systems to settle into a fixed, nonrandom state in the limit of large system size. Typically limit shapes arise due to a combination of entropic considerations and energy minimization. Several diverse areas where limit shapes have been shown to arise are in the theory of random graphs with subgraph density constraints, random tilings with imposed boundary conditions, and random configurational models such as "square ice." These examples are all studied through variational formulations. By studying their common features the PI hopes to learn more about the limit shape phenomenon in general.

水如何变成冰？铁原子的集合如何随温度变化而获得或失去磁性？ “统计力学”是数学和物理学的分支，它涉及理解此类系统：由大量相同对象组成的系统，在本地进行交互。统计力学的主要目标是通过数学方法来描述大规模行为和相变。最有趣，最重要的行为类型之一是，当外力（例如强加的边界条件或其他约束）导致所得合奏中的非均匀性时。一个简单的例子是在重力引起的压力梯度下的水/冰过渡。主要研究者建议在各种基本环境中研究这种行为的数学模型，以期理解它们的共同特征，并能够预测其他潜在更复杂的系统中的相似行为。概率中极限形状的概念描述了大型随机系统的特性，可以在较大的系统大小的极限下定居在固定的，非兰代的状态。通常，由于熵考虑和能量最小化的结合而产生限制形状。已经显示出极限形状出现的几个不同区域是在具有子图密度约束的随机图理论中，随机的边界条件和随机构型模型（例如“平方冰”）。这些示例均通过各种配方研究。通过研究其共同特征，PI希望更多地了解极限形状现象。