Geometry and Integrability of Random Processes

概率论的两个主要目标是解决大型复杂系统如何工作的问题，并确定其演化的几何形状。概率模型广泛应用于生物学、统计物理学、量子力学和机器学习等领域。例子包括癌症生长模型、疾病在人群中的传播、亚原子粒子的支配原理、黑洞、神经网络等。这个项目的目的是理解这些例子代表的一系列模型的几何和内在属性。该项目旨在基于研究者开发的工具解决这些领域的开放性问题。多米诺骨牌图、随机矩阵和随机六顶点模型是统计物理领域的热门领域，而Liouville共形场理论（LCFT）和最优输运理论在量子力学和机器学习领域获得了极大的关注。这个项目围绕着这些领域的问题，旨在获得关于它们的几何和可积性的新见解。具体而言，本项目拟：(1)寻找包括KPZ不动点、随机矩阵的边谱、多米诺骨牌平铺在内的kardar - paris - zhang （KPZ）普域类中模型的迭代对数规律和分形维数规律；(2)建立一个统一的框架，用于研究相互作用粒子系统的力矩公式和包括随机六顶点模型在内的顶点模型；(3)从规范理论严格证明了LCFT共形块和配分函数的模变换性质；(4)研究了熵正则化最优输运在正则化消失时的收敛性。通过融合不同领域的思想，包括聚合物几何、表示理论、黎曼-希尔伯特技术、量子群和凸几何，研究者旨在解决其他方法难以解决的问题。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。

Two major goals of probability theory are to address the question of how large complex systems work and to identify the geometry of their evolution. Probabilistic models are widespread in fields like biology, statistical physics, quantum mechanics, and machine learning. Examples include models of cancer growth, spread of disease in population, governing principles of subatomic particles, black holes, neural networks, etc. The purpose of this project is to understand the geometry and intrinsic properties of a string of models that are representatives of these examples. The project aims to resolve open questions in those fields based on tools that the investigator has developed. Domino tilings, random matrices, and stochastic six vertex models are areas of intense interest in the field of statistical physics, while Liouville conformal field theory (LCFT) and theory of optimal transport have gained immense attention in the fields of quantum mechanics and machine learning. This project revolves around questions in those areas and aims to acquire new insights about their geometry and integrability. In particular, this project plans to: (1) find laws of iterated logarithms and fractal dimension of models in the Kardar-Parisi-Zhang (KPZ) universality class including the KPZ fixed point, edge spectrum of random matrices, and domino tilings; (2) build a unified framework for studying the moment formulas of interacting particle systems and vertex models including the stochastic six vertex model; (3) rigorously prove modular transformation properties of conformal blocks of LCFT and partition functions from gauge theory; and (4) study the convergence of entropically regularized optimal transport to optimal transport when the regularization vanishes. By intermingling ideas from various fields including geometry of polymers, representation theory, Riemann-Hilbert techniques, quantum groups, and convex geometry, the investigator aims to resolve questions that were hard to tackle with other methods.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.