Exactly Solvable Stochastic Systems: Connections and Universality

精确可解的随机系统：联系和普遍性

• 批准号: 1949820
• 负责人: Vadim Gorin
• 金额: $ 18.24万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1949820&HistoricalAwards=false
• 关键词: Exactly; Solvable; Stochastic; Systems; Connections

This project deals with large random systems, which can be typically described as collections of particles with certain prescribed rules of interactions. Examples are given by random stepped surfaces in three-dimensional space, random sorting networks, six-vertex model (also known as the square ice and used to model a thin planar layer of molecules of water). A distinguishing feature of the studied systems is the presence of numerous exact formulas, describing the probabilistic characteristics of the systems. The central questions concern asymptotic properties of the systems of growing sizes, and the research aims at exact rather than qualitative answers.The project aims at two kinds of results. First, the new types of asymptotic behavior lead to discoveries of connections between stochastic systems of different origins, such as lattice models of two-dimensional statistical mechanics, random matrices, stochastic partial differential equations, and probability measures of asymptotic representation theory. Second, the development of robust methods leads to the extensions of the results from the exactly solvable cases to much wider universality classes, thus, justifying the use of these cases for the extensive peculiar predictions, potentially reaching the real-world applications. In particular, during the project the PI will investigate the transition between Gaussian Free Field and Gaussian fields solving hyperbolic SPDEs in the six-vertex model, connections between random sorting networks and eigenvalue distributions of random matrices, and the link between matrix elements of powers of random Hermitian matrices and the finite-time distribution of the Kardar-Parisi-Zhang SPDE.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.

这个项目涉及大型随机系统，它通常可以被描述为具有某些规定的相互作用规则的粒子集合。给出的例子有三维空间中的随机阶梯表面、随机排序网络、六顶点模型(也称为正方形冰，用于模拟薄薄的平面水分子层)。所研究系统的一个显著特点是存在许多描述系统概率特征的精确公式。中心问题涉及增长规模系统的渐近性质，研究的目标是准确的答案而不是定性的答案。首先，新类型的渐近行为导致了不同来源的随机系统之间的联系的发现，例如二维统计力学的格子模型、随机矩阵、随机偏微分方程组和渐近表示理论的概率度量。第二，稳健方法的发展使得结果从精确可解的情形扩展到更广泛的普适性类，从而证明了这些情形用于广泛的特殊预测是合理的，并有可能达到现实世界的应用。特别是，在项目期间，PI将调查高斯自由场和高斯场之间的转换，在六顶点模型中求解双曲SPDEs，随机排序网络和随机矩阵的特征值分布之间的联系，以及随机厄米特矩阵的幂的矩阵元素和Kardar-Parisi-Zhang SPDE的有限时间分布之间的联系。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Absorbing time asymptotics in the oriented swap process

定向交换过程中的吸收时间渐近

• DOI: 10.1214/21-aap1695
• 发表时间: 2022
• 期刊: The Annals of Applied Probability
• 作者: Bufetov, Alexey;Gorin, Vadim;Romik, Dan
• 通讯作者: Romik, Dan

Matrix Addition and the Dunkl Transform at High Temperature

高温下的矩阵加法和 Dunkl 变换

• DOI: 10.1007/s00220-022-04411-z
• 发表时间: 2022
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Benaych-Georges, Florent;Cuenca, Cesar;Gorin, Vadim
• 通讯作者: Gorin, Vadim

Gaussian unitary ensemble in random lozenge tilings

随机菱形平铺中的高斯酉系综

• DOI: 10.1007/s00440-022-01168-3
• 发表时间: 2022
• 期刊: Probability Theory and Related Fields
• 影响因子: 2
• 作者: Aggarwal, Amol;Gorin, Vadim
• 通讯作者: Gorin, Vadim

Shift‐invariance for vertex models and polymers

顶点模型和聚合物的平移不变性

• DOI: 10.1112/plms.12427
• 发表时间: 2022
• 期刊: Proceedings of the London Mathematical Society
• 影响因子: 1.8
• 作者: Borodin, Alexei;Gorin, Vadim;Wheeler, Michael
• 通讯作者: Wheeler, Michael

Cointegration in large VARs

• DOI: 10.1214/21-aos2164
• 发表时间: 2020-06
• 期刊: The Annals of Statistics
• 作者: A. Bykhovskaya;V. Gorin