Large Random Matrices and Determinantal Random Point Fields

大型随机矩阵和行列式随机点域

• 批准号: 0103948
• 负责人: Alexander Soshnikov
• 金额: $ 8.6万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0103948&HistoricalAwards=false
• 关键词: Large; Random; Matrices; Determinantal; Point

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.

主要研究人员将致力于随机矩阵理论和行列式随机点场中的几个问题。主要研究了大型随机矩阵特征值的统计性质，特别是普适性猜想。在以前关于某些Wigner矩阵最大特征值的工作的基础上，他希望将他的结果推广到更广泛的Wigner矩阵，并证明样本协方差矩阵的类似结果。他还建议使用重整化群方法来研究大部分光谱中的普适性。该项目的另一个重点是关于行列式随机点场。目的是找到(重定标度)线性统计的中心极限定理类型结果的充分一般条件，并研究平移不变随机点场的遍历性质。建议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(重核能级统计)、固体物理(模拟小金属粒子和量子点的输运性质)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卓有成效的联系，该领域的重要性增加了。除了建议中指出的结果的各种应用外，主要研究者认为，对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的更好的理解同样重要。