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: 1855458
• 负责人: Vadim Gorin
• 金额: $ 18.24万
• 依托单位: Massachusetts Institute of Technology
• 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 2019-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1855458&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.

该项目涉及大型随机系统，通常可以将其描述为具有某些规定的相互作用规则的粒子集合。三维空间中的随机阶梯表面、随机排序网络、六顶点模型（也称为方形冰，用于模拟水分子的薄平面层）给出了示例。所研究的系统的一个显着特征是存在许多精确的公式，描述了系统的概率特征。中心问题涉及规模不断增长的系统的渐近性质，研究的目标是精确的而不是定性的答案。该项目的目标是两种结果。首先，新型渐近行为导致不同起源的随机系统之间联系的发现，例如二维统计力学的格子模型、随机矩阵、随机偏微分方程和渐近表示理论的概率测度。其次，鲁棒方法的发展导致结果从完全可解的情况扩展到更广泛的普遍性类别，从而证明这些情况用于广泛的特殊预测的合理性，有可能达到现实世界的应用。特别是，在该项目期间，PI 将研究高斯自由场和高斯场之间的过渡，求解六顶点模型中的双曲 SPDE、随机排序网络和随机矩阵特征值分布之间的联系，以及随机 Hermitian 矩阵幂的矩阵元素和 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点和更广泛的影响审查标准进行评估，被认为值得支持。