UM ETheses Collection (澳門大學電子學位論文庫)
 Title

Eigenvalues statistics for restricted trace ensembles
 English Abstract

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Random matrices theory has been extensively studied since the early work of Wigner and Dyson. Random matrices are as effective mathematical reference models for the description of statistical properties in the spectra of complex physical systems. In this dissertation, the author, from a mathematical viewpoint, obtains certainly rigorous results for restricted trace random matrices (RTEs), which bear the same relationship to unconstricted random matrices ensembles as that of the microcanonical ensembles to the canonical ensembles in statistical physics [8]. The dissertation is divided into three parts. First, we briefly introduce three classical random matrices ensembles which are known as the Gaussian unitary ensemble (GUE), the Gaussian orthogonal ensemble (GOE) and the Gaussian symplectic ensemble (GSE). The corresponding probability density functions (p.d.f’s.) of the eigenvalues are presented. Analogously, we review the Wishart (or Laguerre) random matrices ensembles. Based on such background, we can naturally obtain the restricted trace ensembles, and their corresponding joint p.d.fs., especially the restricted trace Laguerre ensembles. After introducing the definition of the correlation functions of the eigenvalues for random matrices ensembles, we give a survey on some known results for the correlation functions, including the sine kernel, Airy kernel and Bessel kernel. Secondly, we consider the fixed and bounded trace Gaussian orthogonal, unitary and symplectic ensembles. In case of the Gaussian ensembles, we prove the universal limits of correlation functions at zero and at the edge of the spectrum edge. The key idea is inspired by [43]. In addition, by using the universal result in the bulk for the fixed trace Gaussian unitary ensemble, which has been obtained by G¨otze and Gordin [42], the universal limits of correlation functions for the bounded trace Gaussian unitary ensembles are obtained. Thirdly, we work on the fixed (bounded) trace unitary ensemble, naturally appears in quantum information and quantum chaos. We focus on correlation functions of Schmidt eigenvalues for the model and prove universal limits of the correlation functions in the bulk and also at the soft and hard edges of the spectrum, as that of the LUE. In the bulk of the spectrum, we make use of the asymptotic behavior of Laguerre polynomials in the complex plane, as developed in [97]. Further we consider the bounded trace LUE and obtain the same universal limits. On the other hand, for finite N, we compute the distribution of the smallest eigenvalue of fixed trace Laguerre ensemble. Furthermore, we found, based on the exact result obtained, a limit distribution, under a certain large N scaling. This turns out tobe the same as the smallest eigenvalue distribution of the classical Laguerre ensembles without the fixed trace constraint. This suggests that in a broad sense global constraint does not influence local correlations at least in the infinite matrix limit. Consequently, by tracing out the environmental degrees of freedom, we obtain the determination of the smallest eigenvalue distribution of the reduced density matrix for a bipartite quantum system of unequal dimensions.
 Issue date

2010.
 Author

Zhou, Da Sheng
 Faculty

Faculty of Science and Technology
 Department

Department of Mathematics
 Degree

Ph.D.
 Subject

Random matrices
 Supervisor

Qian, Tao
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 1/F Zone C
 Library URL
 991005552259706306