Applications of Random Matrices in Physics

Applications of Random Matrices in Physics
Title Applications of Random Matrices in Physics PDF eBook
Author Édouard Brezin
Publisher Springer Science & Business Media
Pages 519
Release 2006-07-03
Genre Science
ISBN 140204531X

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Random matrices are widely and successfully used in physics for almost 60-70 years, beginning with the works of Dyson and Wigner. Although it is an old subject, it is constantly developing into new areas of physics and mathematics. It constitutes now a part of the general culture of a theoretical physicist. Mathematical methods inspired by random matrix theory become more powerful, sophisticated and enjoy rapidly growing applications in physics. Recent examples include the calculation of universal correlations in the mesoscopic system, new applications in disordered and quantum chaotic systems, in combinatorial and growth models, as well as the recent breakthrough, due to the matrix models, in two dimensional gravity and string theory and the non-abelian gauge theories. The book consists of the lectures of the leading specialists and covers rather systematically many of these topics. It can be useful to the specialists in various subjects using random matrices, from PhD students to confirmed scientists.

Products of Random Matrices with Applications to Schrödinger Operators

Products of Random Matrices with Applications to Schrödinger Operators
Title Products of Random Matrices with Applications to Schrödinger Operators PDF eBook
Author P. Bougerol
Publisher Springer Science & Business Media
Pages 290
Release 2012-12-06
Genre Mathematics
ISBN 1468491725

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CHAPTER I THE DETERMINISTIC SCHRODINGER OPERATOR 187 1. The difference equation. Hyperbolic structures 187 2. Self adjointness of H. Spectral properties . 190 3. Slowly increasing generalized eigenfunctions 195 4. Approximations of the spectral measure 196 200 5. The pure point spectrum. A criterion 6. Singularity of the spectrum 202 CHAPTER II ERGODIC SCHRÖDINGER OPERATORS 205 1. Definition and examples 205 2. General spectral properties 206 3. The Lyapunov exponent in the general ergodie case 209 4. The Lyapunov exponent in the independent eas e 211 5. Absence of absolutely continuous spectrum 221 224 6. Distribution of states. Thouless formula 232 7. The pure point spectrum. Kotani's criterion 8. Asymptotic properties of the conductance in 234 the disordered wire CHAPTER III THE PURE POINT SPECTRUM 237 238 1. The pure point spectrum. First proof 240 2. The Laplace transform on SI(2,JR) 247 3. The pure point spectrum. Second proof 250 4. The density of states CHAPTER IV SCHRÖDINGER OPERATORS IN A STRIP 2';3 1. The deterministic Schrödinger operator in 253 a strip 259 2. Ergodie Schrödinger operators in a strip 3. Lyapunov exponents in the independent case. 262 The pure point spectrum (first proof) 267 4. The Laplace transform on Sp(~,JR) 272 5. The pure point spectrum, second proof vii APPENDIX 275 BIBLIOGRAPHY 277 viii PREFACE This book presents two elosely related series of leetures. Part A, due to P.

Spectral Theory Of Large Dimensional Random Matrices And Its Applications To Wireless Communications And Finance Statistics: Random Matrix Theory And Its Applications

Spectral Theory Of Large Dimensional Random Matrices And Its Applications To Wireless Communications And Finance Statistics: Random Matrix Theory And Its Applications
Title Spectral Theory Of Large Dimensional Random Matrices And Its Applications To Wireless Communications And Finance Statistics: Random Matrix Theory And Its Applications PDF eBook
Author Zhaoben Fang
Publisher World Scientific
Pages 233
Release 2014-01-24
Genre Mathematics
ISBN 9814579076

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The book contains three parts: Spectral theory of large dimensional random matrices; Applications to wireless communications; and Applications to finance. In the first part, we introduce some basic theorems of spectral analysis of large dimensional random matrices that are obtained under finite moment conditions, such as the limiting spectral distributions of Wigner matrix and that of large dimensional sample covariance matrix, limits of extreme eigenvalues, and the central limit theorems for linear spectral statistics. In the second part, we introduce some basic examples of applications of random matrix theory to wireless communications and in the third part, we present some examples of Applications to statistical finance.

Introduction to Random Matrices

Introduction to Random Matrices
Title Introduction to Random Matrices PDF eBook
Author Giacomo Livan
Publisher Springer
Pages 122
Release 2018-01-16
Genre Science
ISBN 3319708856

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Modern developments of Random Matrix Theory as well as pedagogical approaches to the standard core of the discipline are surprisingly hard to find in a well-organized, readable and user-friendly fashion. This slim and agile book, written in a pedagogical and hands-on style, without sacrificing formal rigor fills this gap. It brings Ph.D. students in Physics, as well as more senior practitioners, through the standard tools and results on random matrices, with an eye on most recent developments that are not usually covered in introductory texts. The focus is mainly on random matrices with real spectrum.The main guiding threads throughout the book are the Gaussian Ensembles. In particular, Wigner’s semicircle law is derived multiple times to illustrate several techniques (e.g., Coulomb gas approach, replica theory).Most chapters are accompanied by Matlab codes (stored in an online repository) to guide readers through the numerical check of most analytical results.

Random Matrix Theory and Wireless Communications

Random Matrix Theory and Wireless Communications
Title Random Matrix Theory and Wireless Communications PDF eBook
Author Antonia M. Tulino
Publisher Now Publishers Inc
Pages 196
Release 2004
Genre Computers
ISBN 9781933019000

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Random Matrix Theory and Wireless Communications is the first tutorial on random matrices which provides an overview of the theory and brings together in one source the most significant results recently obtained.

Random Matrices and Their Applications

Random Matrices and Their Applications
Title Random Matrices and Their Applications PDF eBook
Author Joel E. Cohen
Publisher American Mathematical Soc.
Pages 376
Release 1986
Genre Mathematics
ISBN 082185044X

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Features twenty-six expository papers on random matrices and products of random matrices. This work reflects both theoretical and applied concerns in fields as diverse as computer science, probability theory, mathematical physics, and population biology.

A Dynamical Approach to Random Matrix Theory

A Dynamical Approach to Random Matrix Theory
Title A Dynamical Approach to Random Matrix Theory PDF eBook
Author László Erdős
Publisher American Mathematical Soc.
Pages 239
Release 2017-08-30
Genre Mathematics
ISBN 1470436485

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A co-publication of the AMS and the Courant Institute of Mathematical Sciences at New York University This book is a concise and self-contained introduction of recent techniques to prove local spectral universality for large random matrices. Random matrix theory is a fast expanding research area, and this book mainly focuses on the methods that the authors participated in developing over the past few years. Many other interesting topics are not included, and neither are several new developments within the framework of these methods. The authors have chosen instead to present key concepts that they believe are the core of these methods and should be relevant for future applications. They keep technicalities to a minimum to make the book accessible to graduate students. With this in mind, they include in this book the basic notions and tools for high-dimensional analysis, such as large deviation, entropy, Dirichlet form, and the logarithmic Sobolev inequality. This manuscript has been developed and continuously improved over the last five years. The authors have taught this material in several regular graduate courses at Harvard, Munich, and Vienna, in addition to various summer schools and short courses. Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University.