Free Probability and Random Matrices

自由概率和随机矩阵

• 批准号: RGPIN-2018-04458
• 负责人: MINGO, James
• 金额: $ 1.31万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=659848
• 关键词: Free; Probability; Random; Matrices

A fundamental problem in many branches of science and engineering is to describe and analyse the distributions of a correlated set of random points, often called a point process. The simplest example of this problem is to independently sample from the same probability distribution. The statistics of this example can be analysed by the laws of classical probability going back to De Moivre. However in many examples the points are correlated in subtle and complicated way and we still want to be able to describe these correlations.******An important example is in the study of correlation matrices. In these matrices the (i,j) entry is given by the correlation between the i-th and j-th variables. By examining the singular numbers of the matrix one can distinguish signal from noise or the absence of correlations between the variables.******In wireless communication theory we have to analyse networks where there are multiple transmitting and receiving antennas. A goal here is to estimate the capacity of the channel and again it is the singular value or eigenvalue distribution that is used.******In quantum information theory it is important to detect entangled states. To solve this problem some mathematical methods have been found called entanglement detectors that announce that a matrix is entangled. One such detector is the partial transpose, for if the partial transpose of a positive matrix fails to be positive then the matrix was entangled. As positivity is an eigenvalue question we are again led to eigenvalue distributions.******Thus the general problem in random matrix theory is to find the eigenvalue distribution of the matrix. Since the entries of the matrix are random the eigenvalues are random. So each instance of the matrix gives n eigenvalues and we get a random set of n points. We shall consider self-adjoint matrices in which case the eigenvalues are all real and so we get a random probability measure on the real line. With random measure we have random moments and these random moments have correlations, skewness, kurtosis, and all higher cumulants.******Typically these higher cumulants are very complicated and thought to be too difficult to analyse. However it was noticed that as the number of points increases the complexity melts away and magically simple and beautiful combinatorial pictures emerge. These are the non-crossing partitions or planar graphs.******The goal of this proposal is to extend my previous work on analysing the asymptotics of these higher cumulants, both combinatorially and in the context of analytic functions. The training component of the proposed research will provide students with an understanding of free probability and random matrices theory which will enable them to pursue careers in academia as well as mathematical finance, wireless communication, and quantum computation.**

科学和工程许多分支中的一个基本问题是描述和分析一组相关的随机点的分布，通常称为点过程。此问题的最简单示例是独立从相同的概率分布中采样。可以通过回到de Moivre的经典概率定律来分析此示例的统计数据。但是，在许多示例中，这些点以微妙而复杂的方式相关，我们仍然希望能够描述这些相关性。******一个重要的例子是在研究相关矩阵中。在这些矩阵中，（i，j）条目由第i-th和j-th变量之间的相关性给出。通过检查矩阵的奇异数字可以将信号与噪声区分开或变量之间没有相关性。******在无线通信理论中，我们必须分析有多次传输和接收天线的网络。这里的目标是估计通道的能力，同样是使用的奇异值或特征值分布。******在量子信息理论中，检测纠缠状态很重要。为了解决此问题，已经发现一些数学方法称为纠缠检测器，宣布矩阵已纠缠。这样的检测器是部分转置，因为如果阳性基质的部分转置未能为阳性，则矩阵纠缠在一起。由于阳性是一个特征值问题，我们再次导致特征值分布。******因此，随机矩阵理论中的一般问题是找到矩阵的特征值分布。由于矩阵的条目是随机的，因此特征值是随机的。因此，矩阵的每个实例都会给出n个特征值，我们会得到一组n个点。我们将考虑自我伴侣矩阵在这种情况下，特征值都是真实的，因此我们在实际线路上获得了一个随机概率度量。通过随机度量，我们有随机的力矩，这些随机的矩具有相关性，偏度，峰度和所有较高的累积剂。******通常，这些较高的累积物非常复杂，并且被认为难以分析。然而，人们注意到，随着积分的数量增加，复杂性融化了，神奇的简单和美丽的组合图片出现了。这些是非交叉的分区或平面图。******该提案的目的是扩展我以前在分析这些较高累积物的渐近物方面的工作，无论是在分析功能上还是在分析功能的背景下。拟议研究的培训部分将为学生提供对自由概率和随机矩阵理论的了解，这将使他们能够从事学术界的职业以及数学金融，无线通信和量子计算。**