Large Random Matrices and Determinantal Random Point Fields

主要研究随机矩阵理论和行列式随机点场中的几个问题。研究的重点是大随机矩阵特征值的统计性质，特别是普适性猜想。在先前关于某些维格纳矩阵的最大特征值的工作的基础上，他希望将他的结果扩展到更广泛的维格纳矩阵类，并证明样本协方差矩阵的类似结果。他还提出利用重整化群方法来研究谱体的普适性。该项目的另一个重点是确定随机点场。目的是为（重新标度的）线性统计的中心极限定理型结果找到充分的一般条件，并研究平移不变随机点场的遍历性质。提出研究的随机矩阵模型来自或应用于多元统计分析（主成分分析）、核物理学（重核能级统计）、固态物理学（小金属粒子和量子点的输运特性建模）和理论计算机科学（计算复杂性、误差统计分析和线性数值算法）。随着数学和物理的许多不同领域，包括组合学、表示理论、算子代数、数论、可积系统、量子混沌、核物理学、统计物理似乎与随机矩阵有着深刻而富有成效的联系，该领域的重要性也在增加。除了在提案中指出的结果的各种应用之外，主要研究者认为，更好地理解随机矩阵中的一些数学现象，特别是特征值局部分布的普适性同样重要。

The principal investigator will work on several problems in random matrix theory and determinantal random point fields. The main emphasis of the research is on statistical properties of the eigenvalues of large random matrices, in particular on the universality conjecture. Building on the previous work on the largest eigenvalues of certain Wigner matrices he expects to extend his results to a wider class of Wigner matrices and prove similar results for sample covariance matrices. He also proposes to study universality in the bulk of the spectrum by using the renormalization group approach. Another foci of the project is concerned with determinantal random point fields. The goal is to find sufficiently general conditions for Central Limit Theorem type results for (rescaled) linear statistics and to study the ergodic properties of translation-invariant random point fields.The random matrix models that are proposed to study come from, or have applications in multivariate statistical analysis (principal component analysis), nuclear physics (statistics of energy levels of heavy nuclei), solid state physics (modelling transport properties of small metallic particles and quantum dots) and theoretical computer science (computational complexity, statistical analysis of errors and linear numerical algorithms). The importance of the field increases as many different areas of mathematics and physics including combinatorics, representation theory, operator algebras, number theory, integrable systems, quantum chaos, nuclear physics, statistical physics appear to have deep and fruitful connections to random matrices. Besides the various applications of the results indicated in the proposal the principal investigator believes that it is equally important to achieve a better understanding of some mathematical phenomena in random matrices, in particular, universality of local distribution of eigenvalues.