Limit Shapes in Probability and Combinatorics

统计力学是数学概率论和理论物理相结合的领域，它的主要目标是理解由许多相互作用的相同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非均匀性。这种非均匀性会导致结构的突然变化，称为空间相变，在这种情况下，系统会破裂成具有非常不同的局部行为的碎片。例如，在压力梯度下，水在同一容器中可以同时具有固相和液相。理解这些相及其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些现象所涉及的一些数学，其目标是基于对其微观相互作用的理解来表征系统的宏观形状及其内部接口。除了发现描述凝聚态物质复杂行为的数学定律之外，这个项目的主要成果之一将是培养数学科学领域的博士生，他们将从概率论和组合学的前沿课题中受益。该研究项目将在几种不同的环境下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（如“方冰”）和相关的可积统计力学模型中。Bethe Ansatz技术虽然在一般情况下非常困难，但可以有效地应用于某些极限情况，以获得数学上严格的极限形状定理，正如本项目计划展示的那样。

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.