Non-asymptotic theory of random matrices

这个项目的重点是一个大尺寸的随机矩阵的谱特性。随机矩阵被用来模拟高维凸性、泛函分析以及物理学和计算机科学中出现的许多问题的典型行为。在所有这些区域中，矩阵的大小，即可用的自由度的数量非常大，但固定。本计画的目的是研究高机率出现的光谱现象，并获得明确的机率界。这里的一个中心问题是评估一个大的随机矩阵是非奇异的概率，并获得非奇异性的定量特征。为此，该项目使用了从经典概率到渐近几何分析和加法组合学等广泛的工具。随机矩阵理论目前正处于快速发展时期。在很大程度上，这是由于在计算机科学和工程的各种问题所产生的明确的概率估计的需求。其中包括无线通信，其中随机矩阵用于模拟天线和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，这是上述问题所需要的。

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.