Large Random Matrices and Random Point Processes

主要研究随机矩阵理论和随机点过程中的几个问题。研究的重点是大随机矩阵的特征值的统计性质，随机矩阵集合中出现的一类重要的随机点过程（即行列式和非定式随机点过程）的基本问题，以及随机矩阵技术在随机增长模型、表示理论和组合学中的应用。随机矩阵理论的主要研究者特别感兴趣的问题包括大样本协方差矩阵谱的局部统计性质、随机图邻接矩阵的谱性质以及具有重尾边缘分布的随机矩阵谱边缘的泊松统计。首席研究员希望在这项研究中与理论物理学家Yan Fyodorov和组合学家Benny Sudakov合作。随机点过程中的问题包括平移不变pfaffian随机点过程的遍历性质的研究，以及确定性和pfaffian系综中Janossy密度的分析。提出研究的随机矩阵模型和随机点过程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（小金属粒子和量子点的输运性质建模），量子混沌（强混沌经典系统的量子类似物的光谱性质）和理论计算机科学(计算复杂性，误差统计分析与线性数值算法)。随着数学和物理的许多不同领域（包括组合学、表示理论、数论、可积系统、随机增长模型、量子引力）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。

0405864Soshnikov The principal investigator works on several problems in random matrix theory and random point processes. The main emphasis of the research is on the statistical properties of the eigenvalues of large random matrices and fundamental questions about an important class of random point processes (namely determinantal and pfaffian random point processes) appearing in random matrix ensembles and applications of random matrix technique in random growth models, representation theory and combinatorics. The problems of a special interest to the principal investigator in random matrix theory include local statistical properties of the spectrum of large sample covariance matrices, spectral properties of the adjacency matrices of random graphs and Poisson statistics at the edge of the spectrum for random matrices with heavy tails of marginal distributions. The principal investigator expects to collaborate in this research with a theoretical physicist Yan Fyodorov and a combinatorialist Benny Sudakov. The problems in random point processes include, among others, the study of the ergodic properties of the translation-invariant pfaffian random point processes and the analysis of the Janossy densities in determinantal and pfaffian ensembles. The random matrix models and random point processes 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 (modeling transport properties of small metallic particles and quantum dots), quantum chaos (spectral properties of the quantum analogues of strongly chaotic classical systems) 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, number theory, integrable systems, random growth models, quantum gravity appear to have deep and fruitful connections to random matrices.