Large Random Matrices and Random Point Processes

0405864Soshnikov主要研究工作在随机矩阵理论和随机点过程的几个问题。研究的主要重点是大型随机矩阵特征值的统计性质和随机矩阵系综中一类重要的随机点过程（即行列式和pf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、随机图的邻接矩阵的谱性质以及边缘分布具有重尾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主要包括：研究不变pf随机点过程的遍历性质，分析行列式系综和pf随机点过程系综中的Janossy密度。 随机矩阵模型和随机点过程都是从多元统计分析中产生的，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），固体物理学（模拟小金属颗粒和量子点的输运性质），量子混沌（强混沌经典系统的量子类似物的光谱特性）和理论计算机科学（计算复杂性，误差的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，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。

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.