Large Random Matrices and Determinantal Random Point Fields

主要研究者将研究随机矩阵理论和行列式随机点场的几个问题。 研究的重点是大型随机矩阵特征值的统计性质，特别是普适性猜想。 建立在以前的工作上的最大特征值的某些维格纳矩阵，他预计将他的结果扩展到更广泛的一类维格纳矩阵，并证明类似的结果样本协方差矩阵。 他还建议研究普遍性的大部分频谱使用重整化群的方法。 该项目的另一个焦点是与行列式随机点场有关。 目标是找到中心极限定理类型结果的充分一般条件，本文研究的随机点场的遍历性，所研究的随机矩阵模型来自于多元统计分析，或者在多元统计分析中有应用主成分分析（principal component analysis），核物理（重核能级统计），固态物理学（模拟小金属粒子和量子点的传输特性）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）。 该领域的重要性随着数学和物理学的许多不同领域而增加，包括组合学，表示论，算子代数，数论，可积系统，量子混沌，核物理，统计物理似乎与随机矩阵有着深刻而富有成效的联系。 除了各种应用的建议中指出的结果的主要研究者认为，它是同样重要的，以实现更好地理解随机矩阵中的一些数学现象，特别是，普遍性的本地分布的特征值。

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.