Colored Stochastic Vertex Models

该项目的目标是复杂概率系统的大时间和大尺度分析。其中许多是为了模拟物理或生物过程的许多组成部分，如细菌生长，燃烧前沿，原子水平上的非常冷的气体，等等。预测这样一个系统在大时间内的行为是非常具有挑战性的，对于数学分析，人们需要依赖于手头模型的某些特殊结构。这种结构起源于代数和表示理论，它最终允许人们以非常精细的分辨率访问大时间和尺度特征。由此产生的概率定律往往出现在更大范围的概率系统中，这可以通过数值和物理实验观察到。这个项目是关于可以用代数方法分析的随机系统；这样的系统通常被称为精确可解或可积。总体目标是发展与任意阶量子仿射群相关的随机顶点模型理论。一方面，这类模型是非常通用的，它统一了许多以前研究过的可积概率模型（如ASEP、随机环境中的定向聚合物和量子玻色子系统），也增加了许多新的模型（如多种或有色排斥过程）。另一方面，可用于这些模型的表示理论框架使得将先前的分析扩展到远远超出当前已知的范围成为可能，更重要的是，发现新的定性和定量的普遍现象。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The goal of the project is large time and scale analysis of complex probabilistic systems. Many of those are designed to model physical or biological processes with numerous components, such as bacterial growth, combustion fronts, very cold gases at the atomic level, and so on. Predicting the behavior of such a system at large times is very challenging, and for mathematical analysis one needs to rely on certain special structure of the model at hand. This structure originates in algebra and representation theory, and it ultimately allows one to access large time and scale characteristics at a very fine resolution. The resulting probabilistic laws tend to be occurring in a substantially wider range of probabilistic systems, which can be observed through numerical and physical experiments. This project is about stochastic systems that can be analyzed by essentially algebraic methods; such systems are often call exactly solvable or integrable. The overarching goal is to develop a theory of stochastic vertex models related to quantum affine groups of arbitrary ranks. On one hand, this class of models is very general, uniting many previously studied integrable probabilistic models (such as ASEP, directed polymers in random environment, and quantum Boson systems), as well as adding many new ones (such as multi-species or colored exclusion processes). On the other hand, the representation theoretic framework available for these models makes it feasible to extend previous analysis far beyond what is currently known and, more importantly, discover new qualitative and quantitative universal phenomena.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.