Exactly Solvable Stochastic Systems: Connections and Universality

这个项目处理大型随机系统，通常可以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）给出了例子。所研究系统的一个显著特征是存在许多精确公式，描述系统的概率特征。中心问题是关于规模不断增长的系统的渐近性质，研究的目的是精确的而不是定性的答案。该项目旨在获得两种结果。首先，新类型的渐近行为导致发现不同起源的随机系统之间的联系，如二维统计力学的晶格模型、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，稳健方法的发展导致结果从精确可解的情况扩展到更广泛的普适性类别，因此，证明这些情况的使用广泛的特殊预测，潜在地达到现实世界的应用。特别是，在项目期间，PI将研究在六顶点模型中求解双曲SPDE的高斯自由场和高斯场之间的转换，随机排序网络与随机矩阵的特征值分布之间的联系，以及随机埃米特矩阵的幂矩阵元素与kardar - paris - zhang SPDE的有限时间分布之间的联系。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。

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.