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，并确定它们演化的几何形状。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。例子包括癌症的增长，疾病在人口中的传播，亚原子粒子，黑洞，神经网络等的管理原则的模型，这个项目的目的是要了解的几何形状和内在属性的一串模型，这些例子的代表。该项目旨在利用调查员开发的工具解决这些领域的未决问题。多米诺骨牌，随机矩阵和随机六顶点模型是统计物理领域的热门领域，而刘维尔共形场论（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。该项目围绕这些领域的问题，旨在获得有关其几何和可积性的新见解。具体而言，本项目计划：（1）在Kardar-Parisi-Zhang（KPZ）普适类中寻找模型（包括KPZ不动点、随机矩阵的边谱和多米诺平铺）的迭代矩和分维规律，（2）建立研究相互作用粒子系统和顶点模型（包括随机六顶点模型）的矩公式的统一框架;（3）严格证明了LCFT的共形块和配分函数的模变换性质;（4）研究了当正则化为零时熵正则化的最优输运到最优输运的收敛性。该研究员将聚合物几何学、表象理论、黎曼-希尔伯特技术、量子群和凸几何等各个领域的思想融合在一起，旨在解决其他方法难以解决的问题。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。

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.