Limit Shapes from a Combinatorial Viewpoint

水是如何变成冰的？铁原子的集合如何随着温度的变化而获得或失去磁性？“统计力学”是数学和物理学的一个分支，研究如何理解这样的系统：由大量相同的物体组成的局部相互作用的系统。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，导致所产生的整体中的非均质性。一个简单的例子是重力引起的压力梯度下的水/冰转变。主要研究人员建议在各种基本设置下研究这种行为的数学模型，以期了解它们的共同特征，并能够预测其他潜在更复杂的系统的类似行为。概率极限形状的概念描述了大型随机系统在大系统规模限制下稳定在固定的非随机状态的性质。通常情况下，极限形状的出现是由于熵考虑和能量最小化的综合考虑。已经证明出现极限形状的几个不同领域是具有子图密度约束的随机图理论、具有强制边界条件的随机平铺以及随机构形模型，例如“正方形冰”。这些例子都是通过变分公式来研究的。通过研究它们的共同特征，PI希望从总体上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.