隨機(jī)矩陣乘積特征根的局部統(tǒng)計(jì)性質(zhì)
發(fā)布時(shí)間：2018-01-31 03:38
本文關(guān)鍵詞： 特征根 Ginibre矩陣 局部統(tǒng)計(jì)性質(zhì) 順序統(tǒng)計(jì)量 隨機(jī)矩陣乘積 截?cái)嘤暇仃?/strong> 出處：《哈爾濱工業(yè)大學(xué)》2017年博士論文 論文類型：學(xué)位論文
【摘要】：獨(dú)立隨機(jī)矩陣乘積的研究可以追溯到Furstenberg和Kesten的先驅(qū)性工作,其在動(dòng)力系統(tǒng)的Lyapunove指數(shù)、薛定諤算子理論、無線通信等方面有重要應(yīng)用。隨機(jī)矩陣乘積研究工作的核心問題是:當(dāng)矩陣維數(shù)趨于無窮時(shí)對(duì)特征根與奇異值全局和局部統(tǒng)計(jì)性質(zhì)的描述。近來,隨機(jī)矩陣乘積的經(jīng)驗(yàn)譜分布(全局性質(zhì))的漸近性被通過自由概率論提供的工具導(dǎo)出。而特征根的局部統(tǒng)計(jì)性質(zhì)并不能通過這類方法得到,因而還有待研究。幸而最近的一系列論文給出了幾類特殊隨機(jī)矩陣乘積(獨(dú)立復(fù)Ginibre矩陣及其逆的乘積、獨(dú)立截?cái)嘤暇仃嚨某朔e、前面兩類矩陣的混合乘積以及一般的統(tǒng)計(jì)各向同性隨機(jī)矩陣的乘積)的特征根的聯(lián)合概率密度函數(shù)。并證明了這些乘積隨機(jī)矩陣的特征根構(gòu)成行列式點(diǎn)過程,這為研究特征根的局部統(tǒng)計(jì)性質(zhì)提供了基礎(chǔ)。本文研究特征根的局部統(tǒng)計(jì)性質(zhì),主要包含以下幾個(gè)部分:首先,本文運(yùn)用鞍點(diǎn)方法給出了一類帶奇點(diǎn)的多元復(fù)積分的漸近性。這類積分出現(xiàn)在了乘積隨機(jī)矩陣的關(guān)聯(lián)核函數(shù)的積分表示中,因此可用于研究乘積隨機(jī)矩陣特征根的局部統(tǒng)計(jì)性質(zhì)。其次,m個(gè)獨(dú)立誘導(dǎo)復(fù)Ginibre矩陣及其逆的乘積的特征根構(gòu)成行列式點(diǎn)過程。其關(guān)聯(lián)核函數(shù)可表示為權(quán)函數(shù)部分與部分和部分的乘積。本文應(yīng)用鞍點(diǎn)方法研究了權(quán)函數(shù)部分的漸近性,并將部分和部分表示為帶奇點(diǎn)的多元復(fù)積分。并利用上面建立的這類積分的漸近性,我們發(fā)現(xiàn)這類隨機(jī)矩陣乘積的特征根的關(guān)聯(lián)函數(shù)不論是在極限經(jīng)驗(yàn)譜分布支撐的內(nèi)部還是在其邊界上的局部統(tǒng)計(jì)性質(zhì)都與單個(gè)復(fù)Ginibre矩陣相同。同時(shí)本文還研究了 m個(gè)獨(dú)立截?cái)嘤暇仃嚨某朔e以及其與有限個(gè)誘導(dǎo)復(fù)Ginibre矩陣的混合乘積,并得到了其特征根的平均經(jīng)驗(yàn)譜分布的極限與局部統(tǒng)計(jì)性質(zhì)。最后,將最優(yōu)傳輸理論運(yùn)用于研究mN個(gè)獨(dú)立N階導(dǎo)數(shù)型多項(xiàng)式隨機(jī)矩陣乘積的特征根的模的順序統(tǒng)計(jì)量。在特征根模的層結(jié)構(gòu)基礎(chǔ)上我們利用最優(yōu)傳輸理論導(dǎo)出了一個(gè)替代原理。我們利用該替代原理得到了:當(dāng)mN/N隨著N → ∞收斂到τ ∈[0,∞)時(shí),第n大的特征根模的極限分布。并通過研究復(fù)Wishart矩陣的最大特征根的漸近性,初步探討最大特征根(譜半徑)的分布與關(guān)聯(lián)函數(shù)局部統(tǒng)計(jì)性質(zhì)之間的聯(lián)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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