Limit Shapes from a Combinatorial Viewpoint

水是如何变成冰的？铁原子的集合是如何随着温度的变化而获得或失去磁性的？“统计力学”是数学和物理学的分支，它涉及理解这样的系统：由大量相同的物体组成的系统，局部相互作用。统计力学的主要目标是通过数学方法描述大尺度行为和相变。最有趣和最重要的行为类型之一是当外力，如强加的边界条件或其他约束，导致在所得到的合奏非均匀性。一个简单的例子是在重力引起的压力梯度下的水/冰转变。主要研究者建议在各种基本设置中研究这种行为的数学模型，希望能够理解它们的共同特征，并能够预测其他潜在更复杂系统的类似行为。概率极限形状的概念描述了大型随机系统在大型系统大小的限制下进入固定的非随机状态的属性。通常，由于熵考虑和能量最小化的组合，极限形状出现。极限形状出现的几个不同的领域是具有子图密度约束的随机图理论、具有强加边界条件的随机平铺以及诸如“方形冰”的随机构形模型。“这些例子都是通过变分公式研究的。通过研究它们的共同特征，PI希望更多地了解极限形状现象。

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.