Random Matrices and Related Topics

有许多随机增长的系统。火的蔓延、融化的冰块的空腔和细菌菌落的生长都是这类例子中的一部分。这种增长模式有一个共同的特征，即如果一个地点已经是模式的一部分，那么它的邻里也可能是模式的一部分。在这个项目中，我们研究了某些类随机增长系统的一些基本的数学性质。一些随机增长系统的潜在涨落是由随机矩阵理论支配的。我们计划研究随机矩阵普适性的范围和局限性。通过研究具有快速变化势的随机矩阵模型，我们应该对普适性的局限性有更多的了解。另一方面，对环和定向聚合物上的完全不对称简单排斥过程的分析将有助于加深对普适类的理解。我们还计划研究Ary过程和Tracy-Widom的基本性质，以及离散Toeplitz行列式的渐近性。在这些项目中，基础系统的可积/可解结构将发挥关键的技术作用。

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.