Applications of Random Matrix Theory to Analytic Number Theory

本课题探讨随机矩阵理论与解析数论的交叉问题。历史上，第一个领域从数学物理中获得了很大的推动力，并且广泛关注随机选择条目的矩阵中的统计模式；这一领域的思想已经在从流行病学到电信等许多学科中得到了应用。第二个领域利用数学分析来研究整数的性质——例如，研究质数的分布或大整数倾向于因式分解的方式；这些都是数学家长期以来一直感兴趣的自然问题，它们在数据安全方面也有应用。这两个领域首先是由l函数的局部零分布与随机矩阵的特征值分布之间令人惊讶的相似性联系起来的，这在很大程度上仍然是推测性的。l函数是解析数论中感兴趣的某些函数。事实上，这种分布已经在各种各样的情况下被发现，理解为什么这些分布如此普遍是一个重要的问题。更详细地说，该项目由两个相关的部分组成。一是利用算术函数的组合分解研究l函数的零点与随机矩阵的特征值之间的联系；类似的分解可以在函数字段设置中找到，并揭示了相关问题。第二部分是研究紧矩阵群上的某些伪随机漫步在多大程度上与经典随机漫步一样是等分布的；与第二个项目有关的结果最近被用来解决关于某些三角多项式分布的开放性问题。这个项目的两个组成部分共同使用组合表示理论，以及他们对弱相关随机变量序列的研究，这种依赖以算术/组合考虑为特征。

This project explores several problems at the intersection of random matrix theory and analytic number theory. The first of these areas historically draws much of its impetus from mathematical physics and broadly concerns statistical patterns found in matrices where entries have been chosen at random; ideas from this area have found applications in a large number of subjects ranging from epidemiology to telecommunications. The second area makes use of mathematical analysis to study properties of the integers -- for instance to study the distribution of primes or the way large integers tend to factor; these are natural problems which mathematicians have been interested in for a long time and which have applications to data-security. The two areas were first linked by the surprising and still largely conjectural resemblance between the local distribution of zeros of L-functions -- these are certain functions which are of central interest in analytic number theory -- and the distribution of eigenvalues of a random matrix. In fact, distributions of this sort have been found in a diverse range of situations, and it is an important problem to understand why these distributions arise so universally. In more detail, the project consists of two related parts. The first is to study the link between zeros of L-functions and eigenvalues of random matrices by making use of combinatorial decompositions of arithmetic functions; analogous decompositions can be found in a function field setting and have shed light on related problems there. The second part is to investigate the extent to which certain pseudo-random walks on compact matrix groups equidistribute in the same fashion as classical random walks do; results pertaining to this second project have recently been used to resolve open questions about the distribution of certain trigonometric polynomials. The two components of the project share in common the use they make of combinatorial representation theory and also their study of sequences of random variables that are weakly dependent, with a dependence characterized by arithmetic/combinatorial considerations.