Universality Limits, Orthogonal Polynomials and Spaces of Entire Functions

该项目的主要目标是研究在酉情况下建立随机矩阵普适性极限的新技术。一个概率分布被放置在n乘n厄米特矩阵的空间上，导致可能的依赖项-与独立分布项的情况不同。然后研究特征值的m元组的间距的相关性，特别是当m是固定的，并且矩阵的大小n接近无穷大时。在大部分谱中涉及sinc核的约束极限被称为普适极限，因为它与点和基本测度无关。一个主要的目标是证明这种普遍性在最低条件下的基本分布，使用新的技术开发的调查。另一个更具挑战性的目标涉及到类似的问题，在频谱的端点，或在频谱内的点，其中潜在的分布表现出不规则性。普适性极限首先在物理学家尤金维格纳的工作中引起了人们的注意。他试图建立中子从重原子核散射的模型。由于大量的相互作用，尝试概率模型似乎很自然。值得注意的是，随机厄米特矩阵的特征值被证明是一个合适的设置，正交多项式是分析的关键工具。随后，随机矩阵与数论、黎曼zeta函数的零点以及其他主题联系在一起。该项目的成果也应该在其他学科的兴趣随机矩阵出现。也希望研究生和本科生将参与研究。

The main goal of the project is to investigate new techniques for establishing universality limits for random matrices, in the unitary case. A probability distribution is placed on the space of n by n Hermitian matrices, leading to possibly dependent entries - as distinct from the case of independently distributed entries. One then studies the correlation of spacing of m-tuples of eigenvalues, especially when m is fixed, and the size n of the matrix approaches infinity. The conjectured limit, which in the bulk of the spectrum involves the sinc kernel, is called a universality limit, because it is independent of the point and the underlying measure. One main goal is to prove this universality in the bulk under minimal conditions on the underlying distribution, using new techniques developed by the investigator. Another more challenging goal involves analogous questions at the endpoints of the spectrum, or at points inside the spectrum, where the underlying distribution exhibits irregularities.Universality limits first rose to prominence in the work of the physicist Eugene Wigner. He was trying to model scattering of neutrons off heavy nuclei. Because of the large number of interactions, it seemed natural to try a probabilistic model. Remarkably, eigenvalues of random Hermitian matrices turned out to be an appropriate setting, and orthogonal polynomials were a key tool in the analysis. Subsequently random matrices have been connected with number theory, and the zeros of the Riemann zeta function, and also with other topics. The project's outcomes should also be of interest in other disciplines where random matrices arise. It is also hoped that graduate and undergraduate students will become involved in the research.