Random Systems from Symmetric Functions and Vertex Models

现代概率论是纯数学的一个领域，有助于加深对自然和社会现实的理解。该项目研究由现实世界问题引发的复杂随机系统，包括冰和其他凝聚态物质的结构、磁性、量子系统、聚合物、热力学和交通模型。 PI 旨在发现和分析生活在离散空间中的系统，这些系统捕获基本的大规模和长期物理现象，但又是“可积的”，即足够简单以进行精确的数学处理。可积随机系统的分析由来自代数结构的显式公式和各种对称性提供支持。特别令人感兴趣的是依赖时间的不可逆系统和模型，其中的杂质显示出广泛的推测普遍行为。该项目为研究生提供研究机会，并通过组织可积概率暑期学校提供培训活动。该项目围绕新的和已知的可积随机系统，其结构可通过杨-巴克斯特方程或对称函数来访问。研究模型的范围包括一维晶格上具有局部或全局相互作用的相互作用粒子系统（分别是不对称简单排除过程和戴森布朗运动的离散模拟）、随机平铺和其他统计机械系统。目标有三个：（1）发现具有可积结构的新模型（例如戴森布朗运动的q-和麦克唐纳类似物）； （2）在众所周知的模型中找到以前被忽视的对称性（例如具有任意初始数据的粒子系统中的时间反转对称性）； （3）探索新的渐近现象并建立它们的普遍性（例如纯吉布斯态和高斯自由场的非均匀变形）。这些目标将通过寻找和分析矩和相关性的精确公式（借助对称函数和顶点模型）以及使用马尔可夫图开发分布对称性或通过在系统中引入许多非齐次参数来实现。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。

Modern probability theory is an area of pure mathematics instrumental in the growing understanding of natural and social reality. The project studies complex random systems motivated by real-world questions, including the structure of ice and other condensed matter, magnetism, quantum systems, polymers, thermodynamics, and traffic models. The PI aims to discover and analyze systems living in discrete space that capture the essential large-scale and long-time physical phenomena, yet are "integrable", that is, simple enough for exact mathematical treatment. The analysis of integrable random systems is powered by explicit formulas and various symmetries coming from algebraic structures. Of special interest are time-dependent irreversible systems and models with impurities displaying a wide range of conjecturally universal behavior. The project provides research opportunities for graduate students as well as training activities through organization of summer school on integrable probability. The project revolves around new and known integrable stochastic systems whose structure is accessible through the Yang-Baxter equation or symmetric functions. The scope of the studied models includes interacting particle systems on the one-dimensional lattice with local or global interaction (asymmetric simple exclusion process and discrete analogs of the Dyson Brownian motion, respectively), random tilings, and other statistical mechanical systems. The goals are three-fold: (1) discover new models with integrable structure (such as q- and Macdonald analogs of Dyson Brownian motion); (2) find previously overlooked symmetries in well-known models (such as time-reversal symmetries in particle systems with arbitrary initial data); and (3) probe new asymptotic phenomena and establish their universality (such as inhomogeneous deformations of pure Gibbs states and the Gaussian Free Field). The goals will be achieved by finding and analyzing exact formulas for moments and correlations (with the help of symmetric functions and vertex models), and also by developing distributional symmetries using Markov maps, or by introducing many inhomogeneous parameters into the system.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.