Free Probability and Random Matrices

在科学和工程的许多分支中，一个基本问题是描述和分析一组相关随机点的分布，通常称为点过程。这个问题最简单的例子是从相同的概率分布中独立采样。这个例子的统计数据可以用追溯到德·莫弗尔的经典概率定律来分析。然而，在许多例子中，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。一个重要的例子是研究相关矩阵。在这些矩阵中，（i,j）项由第i个和第j个变量之间的相关性给出。通过检查矩阵的奇异数，可以区分信号与噪声或变量之间缺乏相关性。在无线通信理论中，我们必须分析有多个发射和接收天线的网络。这里的目标是估计信道的容量，同样，它是使用的奇异值或特征值分布。在量子信息理论中，纠缠态的检测是非常重要的。为了解决这个问题，人们找到了一些数学方法，称为纠缠检测器，它可以宣布矩阵是纠缠的。一个这样的检测器是偏转置，因为如果一个正矩阵的偏转置不为正，那么这个矩阵就被纠缠了。由于正性是一个特征值问题，我们再次得到特征值分布。因此，随机矩阵理论中的一般问题是求矩阵的特征值分布。因为矩阵的元素是随机的，所以特征值也是随机的。所以矩阵的每个实例给出n个特征值我们得到n个点的随机集合。我们将考虑自伴随矩阵，在这种情况下，特征值都是实数，因此我们得到实数线上的随机概率测度。在随机测量中，我们有随机矩，这些随机矩有相关性，偏度，峰度，和所有更高的累积量。通常，这些较高的累积量非常复杂，被认为难以分析。然而，人们注意到，随着点的数量增加，复杂性消失了，神奇地简单而美丽的组合图出现了。这些是非交叉分割或平面图。本建议的目标是扩展我以前在分析这些高累积量的渐近性方面的工作，无论是组合还是在分析函数的背景下。该研究的培训部分将为学生提供自由概率和随机矩阵理论的理解，这将使他们能够在学术界以及数学金融，无线通信和量子计算领域从事职业。

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.