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Colored Stochastic Vertex Models

彩色随机顶点模型

基本信息

批准号：
1853981
负责人：
Alexei Borodin
金额：
$ 54万
依托单位：
Massachusetts Institute of Technology
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1853981&HistoricalAwards=false
关键词：
Colored Stochastic Vertex Models

项目摘要

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.

该项目的目标是对复杂概率系统进行大规模的时间和规模分析。其中许多旨在模拟具有众多组成部分的物理或生物过程，例如细菌生长、燃烧前沿、原子水平的极冷气体等等。长时间预测这样一个系统的行为非常具有挑战性，并且对于数学分析，需要依赖于现有模型的某些特殊结构。这种结构起源于代数和表示论，它最终允许人们以非常精细的分辨率访问大量的时间和尺度特征。由此产生的概率定律往往出现在更广泛的概率系统中，这可以通过数值和物理实验来观察。该项目是关于可以通过本质上代数方法进行分析的随机系统；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的量子仿射群相关的随机顶点模型理论。一方面，这类模型非常通用，结合了许多先前研究的可积概率模型（例如ASEP、随机环境中的定向聚合物和量子玻色子系统），并添加了许多新模型（例如多物种或颜色排除过程）。另一方面，可用于这些模型的表征理论框架使得将先前的分析扩展到远远超出目前已知的范围成为可能，更重要的是，发现新的定性和定量普遍现象。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。