Integrable Structure of Interacting Particles Systems and Quantum Spin Chains

非对称简单排斥过程(ASEP)是晶格上粒子相互作用的模型。ASEP是研究输运现象的最简单、非平凡的随机模型之一，因为它模拟远离平衡的过程。因此，该模型引起了数学家和物理学家的极大关注。该奖项将支持(1)使用光谱方法探索ASEP的基本可积结构，(2)将ASEP推广到多物种，其中现在有二级、三级等粒子，以及(3)半线上的ASEP。所指的可积结构通常命名为“Bethe Anatz”；在第(3)项中，该项目将支持对限制为半条线所需的新结构进行分析。(这里，Bethe Anatz中熟悉的排列之和被有符号排列之和所取代。)一个密切相关的模型是量子自旋模型，称为海森伯格-伊辛模型。该项目将支持对海森堡-伊辛链中的磁区壁问题的分析。这个项目的一个长期目标是研究这些不同模型中的极限定律及其普适性。高斯分布(常见的钟形曲线)很重要，因为它具有普适性；也就是说，它适用于各种看似无关的问题。所有这些问题的基本共同主题是，所研究的对象具有某种程度的“独立性”；或者用更多的物理术语来说，是不相互作用的(或弱相互作用的)。这些类型的问题在物理和数学上都很容易理解。目前的研究涉及强相互作用的过程；因此，没有令人满意的一般理论，因为存在非相互作用的情况。在寻找新的普遍定律的过程中，随机矩阵理论、相互作用粒子系统及其相关极限定理的思想、方法和结果，特别是Tracy-Widom(TW)分布，在科学和工程中的影响越来越大。自从在随机矩阵理论中首次发现TW分布以来，TW分布已经被证明描述了许多随机系统的性质，包括一大类生长界面和定向聚合物系统的涨落。这些相同的TW分布现在经常用于多变量统计分析，在多变量统计分析中，它们已成为从遗传数据推断种群结构的标准统计工具。它们在涉及信号分析和无线通信的工程中的应用才刚刚开始。这个项目扩大了这些理论研究的范围；因此，将向更广泛的受众提供由此产生的数学结果。

he asymmetric simple exclusion process (ASEP) is a model of interacting particles on a lattice. ASEP is one of the simplest, nontrivial stochastic models in which to study transport phenomena as it models processes far from equilibrium. As such the model has attracted much attention from both mathematicians and physicists. This award will support (1) the exploration of the underlying integrable structure of ASEP using spectral methods, (2) the generalization of ASEP to multi-species where there are now second-class, third-class, etc. particles, and (3) ASEP on the half-line. The integrable structure referred to commonly goes under the name ``Bethe Ansatz''; and in (3) this project will support the analysis of the new structures required in restricting to a half-line. (Here the familiar sum over permutations in Bethe Ansatz is replaced by a sum over signed permutations.) A closely related model is the quantum spin model called the Heisenberg-Ising model. This project will support the analysis of the domain wall problem in the Heisenberg-Ising chain. A long-term goal of this project is the study of limit laws and their universality in these various models.The Gaussian distribution (the familiar bell-shaped curve) is important because of its universality; that is, it applies to a wide variety of seemingly unrelated problems. The underlying common theme for all these problems is the fact that the objects under study have some degree of ``independence''; or stated in more physical terms, are non-interacting (or weakly interacting). These types of problems are well-understood both physically and mathematically. Current research involves processes which are strongly interacting; and as such, there is no satisfactory general theory as there exists for the non-interacting cases. In the search for new universal laws, the ideas, methods and results of random matrix theory, interacting particle systems and their associated limit theorems; in particular, the Tracy-Widom (TW) distributions, have found an ever widening impact in science and engineering. Since their initial discovery in random matrix theory, the TW distributions have been shown to describe the properties of a number of random systems including the fluctuations of a large class of growing interfaces and directed polymer systems. These same TW distributions are now regularly used in multivariate statistical analysis where they have become a standard statistical tool for inferring population structure from genetic data. Their use in engineering involving signal analysis and wireless communications is just beginning. This project extends the scope of these theoretical investigations; and as such, will make available to a wider audience the resulting mathematical results.