Non-asymptotic theory of random matrices

本课题主要研究大型随机矩阵的谱性质。在高维凸性、泛函分析以及物理和计算机科学中出现的许多问题中，随机矩阵被用来模拟典型行为。在所有这些领域中，矩阵的大小，即可用自由度数非常大，但是固定的。这个项目的目的是研究高概率出现的光谱现象，并获得显式的概率界。这里的中心问题之一是评估大型随机矩阵非奇异的概率，并获得非奇异性的定量特征。为此，该项目使用了广泛的工具，从经典概率，到渐近几何分析，以及加性组合学。随机矩阵理论目前正在经历一个快速发展的时期。这在很大程度上是由于在计算机科学和工程的各种问题中出现的对显式概率估计的需求。这包括无线通信，其中使用随机矩阵来模拟天线与地面障碍物之间的相互作用。随机矩阵的研究对于互联网隐私保护也很重要，因为某些类型的随机矩阵成为构建黑客算法的首选工具。该项目将制定明确的定量光谱界限，这是上述问题所必需的。

This project focuses on spectral properties of random matrices of a large size. A random matrix is used to model a typical behavior in many problems arising in high-dimensional convexity, functional analysis, as well as in physics and computer science. In all these areas the size of a matrix, i.e. the available number of degrees of freedom is very large, but fixed. This project aims to investigate the spectral phenomena which arise with high probability, and obtain explicit probability bounds. One of the central questions here is evaluation of the probability that a large random matrix is non-singular, and obtaining quantitative characteristics of non-singularity. To this end, the project brings into play a wide array of tools, ranging from classical probability, to asymptotic geometric analysis, and additive combinatorics.The theory of random matrices is currently undergoing a period of rapid development. In a large part this is due to the demand for explicit probabilistic estimates arising in various problems of computer science and engineering. This includes, among others, the wireless communication, where random matrices are used to model interactions between antennas and obstacles on the ground. The study of random matrices is also of importance for internet privacy protection, since certain kinds of random matrices became tools of choice in constructing hacking algorithms. The project will develop explicit quantitative spectral bounds, which are required for the aforementioned problems.