Random matrices and related models

在单车道的交通情况下，一辆慢车可能会耽误后面一排车。这种情况是随机交互系统的一个例子，其中粒子（汽车）有随机的移动时间，一个粒子的移动会影响一定距离内的其他粒子。火的蔓延，冰融化的空洞，细菌菌落的生长也有类似的特征。在这个项目中，首席研究员研究了一类表示这种随机相互作用的简化模型的一些潜在的基本数学性质。特别是，首席研究员研究粒子运动的随机性如何影响系统的全局特性。首席研究员将研究几个随机系统的波动。它们包括自旋玻璃、随机矩阵和相互作用粒子系统。总体目标是探索由随机矩阵理论和相关模型产生的泛分布函数的范围和局限性，并发现新的泛分布。主要研究了自旋玻璃的2自旋球形Sherrington-Kirkpatrick模型、环上的完全不对称简单不相容过程和Airy过程。主要研究者广泛地利用这些模型的精确可解性来获得精确的渐近结果。黎曼-希尔伯特问题和坐标Bethe ansatz方法是使用的一些工具。

In a one-lane traffic situation, one slow car may delay a whole platoon of cars behind. This situation is an example of randomly interacting systems in which particles (cars) have random amounts of time to move, and one particle's movement influences other particles in some distance. The spread of fire, the cavity of melting ice, and growth of bacterial colonies also have similar features. In this project, the principal investigator studies some of the underlying fundamental mathematical properties for a class of simplified models representing such random interactions. In particular, the principal investigator studies how the randomness of the particles movement impact the global characteristics of the system.The principal investigator will study the fluctuations of several random systems. They include spin glass, random matrices, and interacting particle systems. The overarching goal is to explore the scope and limitation of the universal distribution functions that arise from random matrix theory and related models and also uncover new universal distributions. Specifically, the principal investigator studies 2-spin spherical Sherrington-Kirkpatrick model of spin glass, totally asymmetric simple exclusion process on a ring, and the Airy process. the principal investigator uses the exactly solvable nature of these models extensively to obtain precise asymptotic results. Riemann-Hilbert problems and coordinate Bethe ansatz method are some of the tools used.