随机矩阵乘积特征根的局部统计性质

发布时间：2018-01-31 03:38

  本文关键词： 特征根 Ginibre矩阵 局部统计性质 顺序统计量 随机矩阵乘积 截断酉矩阵　出处：《哈尔滨工业大学》2017年博士论文　论文类型：学位论文

【摘要】：独立随机矩阵乘积的研究可以追溯到Furstenberg和Kesten的先驱性工作,其在动力系统的Lyapunove指数、薛定谔算子理论、无线通信等方面有重要应用。随机矩阵乘积研究工作的核心问题是:当矩阵维数趋于无穷时对特征根与奇异值全局和局部统计性质的描述。近来,随机矩阵乘积的经验谱分布(全局性质)的渐近性被通过自由概率论提供的工具导出。而特征根的局部统计性质并不能通过这类方法得到,因而还有待研究。幸而最近的一系列论文给出了几类特殊随机矩阵乘积(独立复Ginibre矩阵及其逆的乘积、独立截断酉矩阵的乘积、前面两类矩阵的混合乘积以及一般的统计各向同性随机矩阵的乘积)的特征根的联合概率密度函数。并证明了这些乘积随机矩阵的特征根构成行列式点过程,这为研究特征根的局部统计性质提供了基础。本文研究特征根的局部统计性质,主要包含以下几个部分:首先,本文运用鞍点方法给出了一类带奇点的多元复积分的渐近性。这类积分出现在了乘积随机矩阵的关联核函数的积分表示中,因此可用于研究乘积随机矩阵特征根的局部统计性质。其次,m个独立诱导复Ginibre矩阵及其逆的乘积的特征根构成行列式点过程。其关联核函数可表示为权函数部分与部分和部分的乘积。本文应用鞍点方法研究了权函数部分的渐近性,并将部分和部分表示为带奇点的多元复积分。并利用上面建立的这类积分的渐近性,我们发现这类随机矩阵乘积的特征根的关联函数不论是在极限经验谱分布支撑的内部还是在其边界上的局部统计性质都与单个复Ginibre矩阵相同。同时本文还研究了 m个独立截断酉矩阵的乘积以及其与有限个诱导复Ginibre矩阵的混合乘积,并得到了其特征根的平均经验谱分布的极限与局部统计性质。最后,将最优传输理论运用于研究mN个独立N阶导数型多项式随机矩阵乘积的特征根的模的顺序统计量。在特征根模的层结构基础上我们利用最优传输理论导出了一个替代原理。我们利用该替代原理得到了:当mN/N随着N → ∞收敛到τ ∈[0,∞)时,第n大的特征根模的极限分布。并通过研究复Wishart矩阵的最大特征根的渐近性,初步探讨最大特征根(谱半径)的分布与关联函数局部统计性质之间的联系。
[Abstract]:The study of the product of independent random matrices can be traced back to the pioneering work of Furstenberg and Kesten, which is based on the Lyapunove exponent of dynamical systems and the Schrodinger operator theory. The key problem in the research of random matrix product is the description of the global and local statistical properties of the eigenvalue and singular value when the matrix dimension tends to infinity. The asymptotic property of empirical spectral distribution (global property) of the product of random matrix is derived by the tool provided by free probability theory, while the local statistical property of characteristic root can not be obtained by this kind of method. Fortunately, a series of recent papers have given several special random matrix products (product of independent complex Ginibre matrix and its inverse, product of independent truncated unitary matrix). The joint probability density function of the eigenroots of the first two classes of matrices and the product of the general statistical isotropic random matrices) is proved. The eigenroots of these product random matrices form the determinant point process. This provides a basis for studying the local statistical properties of characteristic roots. In this paper, the local statistical properties of characteristic roots are mainly studied in the following parts: first. In this paper, the asymptotic behavior of a class of multivariate complex integrals with singularities is given by means of saddle point method, which appears in the integral representation of the correlation kernel function of the product random matrix. Therefore, it can be used to study the local statistical properties of eigenvalues of product random matrices. The characteristic roots of M independently induced complex Ginibre matrices and their inverse products form determinant point processes. The correlation kernel function can be expressed as the product of the weight function part and the partial sum part. In this paper, the saddle point method is used to study the process of determinant point. The asymptotic behavior of the weight function part is given. The partial and partial representations are expressed as multivariate complex integrals with singularities, and the asymptotic properties of such integrals established above are used. We find that the correlation function of the eigenroot of the product of this kind of random matrix is the same as that of a single complex Ginibre matrix both in the interior supported by the limit empirical spectrum distribution and on its boundary. This paper also studies. The product of m independent truncated unitary matrices and their mixed products with finite number of induced complex Ginibre matrices. The limit and local statistical properties of the average empirical spectrum distribution of its characteristic roots are obtained. The optimal transmission theory is applied to study the order statistics of the modules of eigenroots of M N independent N-order derivative polynomial random matrices. On the basis of the hierarchical structure of characteristic root modules, we derive an optimal transmission theory. We use this alternative principle to get:. When mN/N with N. 鈫掆垶 convergence to 蟿 鈭，

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