Integrable Probability

该项目专注于研究各种概率系统的大时间和大尺度行为。其中许多都是为了模拟不同的自然过程而设计的，如细菌生长、晶体融化、原子水平上的非常冷的气体等。在大时间或大尺度上进行准确的分析通常是非常困难的，该项目集中在具有额外代数结构的模型上，这些模型起源于看似不相关的数学领域。这种结构有助于发现新的现象，这些现象往往是普遍的(即在非常广泛的系统中存在)。因此，主要研究人员使用复杂的数学工具，试图找到新的普遍定律，它扮演着著名而熟悉的钟形曲线定律的角色，并且经常可以通过物理和数值实验观察到。新兴的可积概率领域的目标是准确地识别和分析可解的概率模型。这些模型和结果往往很容易描述，但很难找到，而且它们包含了关于随机过程的广义普适性类的基本信息。该项目的目的是在深代数论和表示论结构之间建立一座桥梁，在另一端建立概率系统，从而允许利用前者来发现和研究后者。在过去五年中引入和发展的麦克唐纳进程框架在这条道路上相当成功，而且还在继续发展。已知的应用包括相互作用的粒子系统，(1+1)和(2+1)维的随机生长界面，随机矩阵和对数气体，以及随机介质中的定向聚合物。该框架现在准备扩展到包括统计力学的可解晶格模型理论，在这个久负盛名的领域提供全新的视角和新的结果。

The project focuses on studying large time and scale behavior for a variety of probabilistic systems. Many of those were designed to model different natural processes such as bacterial growth, crystal melting, very cold gases at atomic levels, etc. An accurate analysis at large times or scales is typically very difficult, and the project concentrates on models with additional algebraic structure that originate in seemingly unrelated areas of mathematics. This structure helps to discover new phenomena that tend to be universal (i.e., present in a very wide range of systems). As a result, using sophisticated mathematical tools, the principal investigator seeks to find new universal laws that play the role of the famous and familiar bell curve law and that can often be observed through physical and numerical experiments.The goal of the emerging field of integrable probability is to identify and analyze exactly solvable probabilistic models. The models and results are often easy to describe, yet difficult to find, and they carry essential information about broad universality classes of stochastic processes. The project aims at developing a bridge between deep algebraic and representation theoretic structures on one end, and probabilistic systems on the other end, that would allow to utilize the former in order to discover and study the latter. The framework of Macdonald processes introduced and developed in the last five years has been quite successful on this path, and it continues to grow. Known applications include interacting particle systems, random growing interfaces in (1+1) and (2+1) dimensions, random matrices and log-gases, and directed polymers in random media. The framework is now poised to expand to include the theory of solvable lattice models of statistical mechanics, offering completely new perspective and new results in this well-established domain.