FRG: Collaborative Research: Integrable Probability

许多现代概率论研究试图理解大而复杂的随机系统的行为（例如，无序介质中的增长、裂缝、湍流或交通流），目的是发展具有预测和统计价值的理论。虽然人们可以尝试在计算机上直接模拟这样的系统，但它们的大小和复杂性往往使这种尝试无果而终。相反，人们可以寻找这样的系统模型，这些模型足够复杂，可以显示所研究的所有现象，但又足够简单，可以进行精确的数学计算来探测这种行为。可积概率是发现和随后分析这些模型背后的理论。该项目旨在统一该领域和各种最近的突破，并在此过程中发现一系列新型可积概率系统，用于分析的新工具，以及新的大规模普遍现象。可积概率论是概率论、数学物理和统计物理与表示理论和可积系统相结合的一个研究领域。可积概率系统具有两个特性：可以写出简洁而精确的公式来表示系统的各种有趣的可观测值的期望；系统的渐近性、可观测性和公式提供了对新现象和普适性类的精确描述（包含的不仅仅是可积的例子）。可积概率系统的发现和分析取决于其潜在的代数结构。这些可积概率系统可以看作是强大对象的投影，其起源在于表征理论和可积系统。在可积概率系统的研究中有丰富的重大突破的历史，包括六顶点模型，Ising模型，以及最近的KPZ通用性类中的某些模型。许多现有结果的核心基本机制是舒尔/麦克唐纳过程（建立在对称多项式结构之上）和量子可积系统（建立在Yang-Baxter方程和Bethe ansatz方程的解之上）。每种机制都产生了突破性的结果，例如最近解决了25年的物理学猜想，即KPZ随机偏微分方程属于KPZ通用性类。直到最近，这两种可积概率的途径相对独立地存在。该项目的目标是创建一个统一的可积概率理论，结合和推广舒尔/麦克唐纳过程和量子可积系统的方法，并补充提取新的可分析模型和发现新的概率或物理现象。

Much of modern probability research seeks to understand the behavior of large and complex random systems (for instance, growth in disordered media, cracking, turbulent fluids, or traffic flow) with an aim towards developing theories with predictive and statistical value. While one can try to directly model such systems on computers, their size and complexity often render such attempts fruitless. Instead, one can look for models of such systems that are complex enough to display all of the phenomena under study, yet simple enough to admit exact mathematical computation to probe that behavior. Integrable probability is the theory behind discovering and subsequently analyzing such models. This project seeks to unify the area and various recent breakthroughs and in so doing discover a host of new types of integrable probability systems, new tools for their analysis, and new large-scale universal phenomena.Integrable probability is an area of research at the interface between probability, mathematical physics, and statistical physics on the one hand, and representation theory and integrable systems on the other. Integrable probabilistic systems are characterized by two properties: It is possible to write down concise and exact formulas for expectations of a variety of interesting observables of the systems; and asymptotics of the systems, observables, and formulas provide access to exact descriptions of new phenomena and universality classes (containing more than just integrable examples). The discovery and analysis of integrable probabilistic systems hinges upon underlying algebraic structure. These integrable probabilistic systems can be viewed as projections of powerful objects whose origins lie in representation theory and integrable systems. There is a rich history of major breakthroughs in the study of integrable probabilistic systems, including the six-vertex model, Ising model, and more recently certain models in the KPZ universality class. The basic mechanisms at the heart of many of these existing results are Schur / Macdonald processes (built off the structure of symmetric polynomials) and quantum integrable systems (built off solutions to the Yang-Baxter equation and the Bethe ansatz). Each mechanism has produced breakthrough results, such as the recent resolution of the 25-year-old physics conjecture that the KPZ stochastic partial differential equation is in the KPZ universality class. Until recently, these two routes to integrable probability have existed relatively separately. The goal of the proposed project is to create a unified theory of integrable probability, combining and generalizing the methods of Schur / Macdonald processes and quantum integrable systems and, complementarily, extracting new analyzable models and uncovering new probabilistic or physical phenomena.