Integrable Systems, Integral Operators, and Probabilistic Models

自然现象是复杂的。数学建模的目的是研究人们在自然界中所见的公认的更简单的版本，并提取准确的结果，这些结果可以帮助解释，有时还可以预测现实世界中的事件。有“非平衡进程”和“均衡进程”，“非平衡进程”是指系统正在演化，而“均衡进程”则是在系统稳定下来后，对系统的行为进行建模。这个项目研究了其中的一个，“伊辛模型”(一个均衡过程)和“非对称简单排斥过程”(一个非平衡过程)。第一种是对铁磁性的最简单(尽管仍然相当复杂)的描述。我们研究的是二维模型，特别是它的热力学性质(其中粒子的数量变得无限大)。磁性和磁性材料在整个技术中无处不在，因此获得它们的准确数学结果是很重要的。非对称简单排斥过程是粒子相互作用现象的最简单模型。准确地说，没有两个人可以同时占据同一地点。该模型在非平衡统计物理、工程和生物系统中有着广泛的应用。一维情形就是我们所研究的(想象一下电子在导线中流动的情形)。主要研究人员和合作者的理论结果已被最近的实验所证实。这个项目将导致对这一过程和统计物理学中相关随机模型的更深入的理解。非对称简单排除过程(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。在一定的标度极限下，ASEP与Kardar-Parisi-Zhang(KPZ)方程密切相关，KPZ方程是一个非线性随机偏微分方程组，有望描述一大类随机生长的界面。Sasamoto、Spohn、Amir、Corwin和Quastel最近的工作，利用Tracy和PI的结果，严格地建立了ASEP和KPZ方程之间的这种联系。这个项目建立在Tracy和PI工作的基础上，并涉及：(I)具有开放边界的ASEP渐近结果的严格算符理论基础；(Ii)对ASEP的离散时间版本的研究；以及(Iii)Weyl群在分类Bethe Ansat可解的相互作用粒子系统中的作用。

Natural phenomena are complex. The purpose of mathematical modeling is to study admittedly simpler versions of what one sees in nature and extract exact results which could help explain, and sometimes predict, events in the real world. There are "nonequlibrium processes", in which the system is evolving, and "equlibrium processes" which model the behavior of the system after it has settled down. This project studies one of each, the "Ising model" (an equlibrium process) and the "asymmetric simple exclusion process" (a nonequlibrium process). The first is the simplest (although still quite complicated) description of ferromagnetism. It is the two-dimensional model that is studied, in particular its thermodynamic properties (in which the number of particles gets infinitely large). Magnetism and magnetic materials are pervasive throughout technology, so it is important to achieve exact mathematical results for them. The asymmetric simple exclusion process is the simplest model of phenomena in which particles interact with each other. Precisely, no two can occupy the same site at the same time. The model has far-ranging applications in nonequilibrium statistical physics, engineering, and biological systems. The one-dimensional case is what is studied (think of electrons flowing in a wire). Theoretical results of the principal investigator and collaborator have been confirmed by recent experiment. This project will result in a deeper understanding of this process and related random models in statistical physics.The asymmetric simple exclusion process (ASEP) is one of the simplest, nontrivial stochastic models in which to study transport phenomena as it models processes far from equilibrium. ASEP is closely related, in a certain scaling limit, to the Kardar-Parisi-Zhang (KPZ) equation, a nonlinear stochastic partial differential equation that is expected to describe a large class of stochastically growing interfaces. Recent work by Sasamoto, Spohn, Amir, Corwin and Quastel, using results of Tracy and the PI, have rigorously established this link between ASEP and the KPZ equation. This project builds on the work of Tracy and the PI and involves: (i) the rigorous operator-theoretic underpinnings of asymptotic results for ASEP with an open boundary; (ii) the study of a discrete time version of ASEP; and (iii) the role of Weyl groups in classifying interacting particle systems solvable by a Bethe Ansatz.