Spectral Properties of Large Random Matrices

主要研究随机矩阵论和概率论中的问题，主要侧重于大型随机矩阵的特征值的统计性质。P.I.打算继续研究Wigner中最大特征值的分布、样本协方差以及相关的随机矩阵集合。所提出的研究与厄米特和实对称随机矩阵的最大特征值的Tracy-Widom分布密切相关，该分布出现在物理、统计和理论计算机科学中的许多重要问题中：晶体形状、排斥过程、随机介质中的定向极化、主成分分析。特别是，P.I.建议在一些重要的随机矩阵集合中建立Tracy-Widom分布。最近，P.I.发现了具有指数尾部元素的随机矩阵的最大特征值的Poisson统计量。值得注意的是，这种矩阵经常出现在金融应用中。P.I.打算继续他对重尾随机矩阵的研究，特别是详细地研究从Tracy-Widom体制到Poisson体制的过渡。项目中研究的随机矩阵模型来自或应用于核物理(重核能级统计)、数理统计(主成分分析)、理论计算机科学(计算复杂性、误差统计分析、线性数值算法)、种群生物学、数学金融和固体物理(模拟小金属粒子和量子点的输运性质)。在过去的十年里，该领域的重要性显著增加，因为许多不同的数学和物理领域，包括组合学、表示论、数论、量子重力、可积系统和随机增长模型，都被证明与随机矩阵理论有着深刻而卓有成效的联系。

The principal investigator works on problems in Random Matrix Theory and Probability Theory with the main emphasis on statistical properties of the eigenvalues of large random matrices. The P.I. intends to continue his studies of the distribution of the largest eigenvalues in Wigner, sample covariance, and related ensembles of random matrices. The proposed research is closely related to the Tracy-Widom distribution of the largest eigenvalues of Hermitian and real symmetric random matrices that appears in many important problems in physics, statistics, and theoretical computer science: crystal shapes, exclusion processes, directed polimers in random media, principal component analysis. In particular, the P.I. proposes to establish Tracy-Widom distribution in some important ensembles of random matrices. Recently, the P.I. discovered Poisson statistics of the largest eigenvalues in random matrices with power-law tailed elements. Remarkably, such matrices very often appear in financial applications. The P.I. intends to continue his studies of heavy-tailed random matrices and, in particular, to study in detail the transition form the Tracy-Widom regime to the Poisson regime. The random matrix models studied in the project come from or have applications in nuclear physics (statistics of energy levels of heavy nuclei), mathematical statistics (principal component analysis), theoretical computer science (computational complexity, statistical analysis of errors, linear numerical algorithms), population biology, mathematical finance, and solid state physics (modeling transport properties of small metallic particles and quantum dots). The importance of the field has increased significantly in the last ten years as many different areas of mathematics and physics including combinatorics, representation theory, number theory, quantum gravity, integrable systems, and random growth models have been shown to possess deep and fruitful connections to random matrix theory.