Limit Shapes in Probability and Combinatorics

概率和组合学中的极限形状

• 批准号: 1612668
• 负责人: Richard Kenyon
• 金额: $ 7.95万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1612668&HistoricalAwards=false
• 关键词: Limit; Shapes; Probability; Combinatorics

The main goal of statistical mechanics, a field at the interface of mathematical probability and theoretical physics, is to understand the collective behavior of systems consisting of many interacting identical particles. 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 system of particles. This non-homogeneity can result in abrupt changes in structure, called spatial phase transitions, where the system breaks into pieces with very different local behaviors. For example under a pressure gradient, water can have both solid and liquid phases in the same container. It is a challenging mathematical problem to understand these phases and their common boundaries. The project will investigate some of the mathematics involved in describing these kinds of phenomena at a generic level, with a goal to characterize the macroscopic shapes of the systems and their internal interfaces based on an understanding of their microscopic interactions. One of the main outcomes of this project, beyond the discovery of mathematical laws describing the complex behavior of condensed matter, will be the training of Ph.D. students in the mathematical sciences who will profit from working on cutting-edge topics in probability theory and combinatorics.The research project will study mathematical models of such limiting-shape behaviors, in several different settings, notably in planar configurational models such as "square ice" and related integrable statistical mechanical models. Bethe Ansatz techniques, although notoriously difficult in general, can be effectively applied in certain limiting situations to obtain mathematically rigorous limit shape theorems, as this project plans to show.

统计力学是数学概率和理论物理交界处的一个领域，其主要目标是理解由许多相互作用的完全相同的粒子组成的系统的集体行为。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，导致所产生的粒子系统中的不均匀。这种非均质性可能导致结构的突然变化，称为空间相变，即系统分裂成具有非常不同的局部行为的碎片。例如，在压力梯度下，水可以在同一容器中同时具有固体和液体。理解这些阶段和它们的共同边界是一个具有挑战性的数学问题。该项目将研究在一般水平上描述这类现象所涉及的一些数学，目的是基于对它们微观相互作用的理解来表征系统及其内部界面的宏观形状。这个项目的主要成果之一，除了发现描述凝聚态复杂行为的数学规律外，还将培训数学科学的博士生，他们将从概率论和组合学的前沿课题中受益。该研究项目将在几个不同的环境中研究这种极限形状行为的数学模型，特别是在平面构型模型中，如“正方形冰”和相关的可积统计力学模型。正如本项目计划展示的那样，Bethe Ansat技术虽然在总体上是出了名的困难，但在某些极限情况下可以有效地应用，以获得数学上严格的极限形状定理。