Random Matrices and Related Topics

有许多随机生长的系统。火灾的蔓延，冰块融化的空洞，细菌菌落的生长都是这类的例子。这样的增长模式有一个共同的特点，即如果一个地点已经成为该模式的一部分，那么它的社区也可能成为该模式的一部分。在这个项目中，我们研究了这类随机增长系统的一些潜在的基本数学性质。一些随机生长系统的基本涨落是由随机矩阵理论控制的。我们计划研究随机矩阵的普适性的范围和限制。通过对具有快速变化势的随机矩阵模型的研究，我们可以更深入地了解通用性的局限性。另一方面，对环和定向聚合物上的完全不对称简单不相容过程的分析将提高我们对普适类的理解。我们还计划研究Airy过程和Tracy-Widom的基本性质，以及离散Toeplitz行列式的渐近性。在项目中，底层系统的可积/可解结构将扮演关键的技术角色。

There are many randomly growing systems. Spreads of fire, the cavity of melting ice cube, and growth of bacterial colonies are some of the examples of this kind. Such growth patterns have the common feature that if one location is already a part of the pattern, then its neighborhoods are also likely to a part of the pattern. In this project, we study some of the underlying fundamental mathematical properties for certain classes of such randomly growing systems. The underlying fluctuations of some of randomly growing systems are governed by the random matrix theory. We plan to study the scope and the limitations of the universality of random matrices. By studying the random matrix models with rapidly changing potential for which the bulk universality is not expected to hold, we should gain more insights on the limitations of the universality. On the other hand, the analysis of the totally asymmetric simple exclusion process on a ring and directed polymers will improve out understanding of the universality class. We also plan to study the fundamental properties of the Airy process and the Tracy-Widom, and also the asymptotics of discrete Toeplitz determinants. In the projects, the integrable/solvable structure of the underlying system will place a key technical role.