Integrable Probability

该项目侧重于研究各种概率系统的大时间和尺度行为。其中许多是为了模拟不同的自然过程而设计的，比如细菌生长、晶体融化、原子水平上的非常冷的气体等等。在大时间或尺度上进行准确的分析通常是非常困难的，该项目集中在具有附加代数结构的模型上，这些模型起源于看似无关的数学领域。这种结构有助于发现趋向于普遍的新现象（即存在于非常广泛的系统中）。因此，利用复杂的数学工具，首席研究员试图找到新的普遍规律，这些规律扮演着著名和熟悉的钟形曲线定律的角色，并且通常可以通过物理和数值实验观察到。可积概率这一新兴领域的目标是识别和分析精确可解的概率模型。模型和结果通常很容易描述，但很难找到，它们携带了关于随机过程的广泛普遍性的基本信息。该项目旨在建立一个桥梁，一端是深度代数和表示理论结构，另一端是概率系统，这将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流程框架在这条道路上取得了相当大的成功，并且还在继续发展。已知的应用包括相互作用粒子系统，（1+1）和（2+1）维度的随机生长界面，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，在这个已建立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.