Random Matrices High Dimensional Phenomena

Author : Gordon Blower
Publisher : Cambridge University Press
Page : 448 pages
File Size : 50,6 Mb
Release : 2009-10-08
Category : Mathematics
ISBN : 0521133122

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Random Matrices: High Dimensional Phenomena by Gordon Blower Pdf

This book focuses on the behavior of large random matrices. Standard results are covered, and the presentation emphasizes elementary operator theory and differential equations, so as to be accessible to graduate students and other non-experts. The introductory chapters review material on Lie groups and probability measures in a style suitable for applications in random matrix theory. Later chapters use modern convexity theory to establish subtle results about the convergence of eigenvalue distributions as the size of the matrices increases. Random matrices are viewed as geometrical objects with large dimension. The book analyzes the concentration of measure phenomenon, which describes how measures behave on geometrical objects with large dimension. To prove such results for random matrices, the book develops the modern theory of optimal transportation and proves the associated functional inequalities involving entropy and information. These include the logarithmic Sobolev inequality, which measures how fast some physical systems converge to equilibrium.

Author : Gordon Blower
Publisher : Cambridge University Press
Page : 448 pages
File Size : 54,8 Mb
Release : 2009-10-08
Category : Mathematics
ISBN : 9781139481953

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Random Matrices: High Dimensional Phenomena by Gordon Blower Pdf

This book focuses on the behaviour of large random matrices. Standard results are covered, and the presentation emphasizes elementary operator theory and differential equations, so as to be accessible to graduate students and other non-experts. The introductory chapters review material on Lie groups and probability measures in a style suitable for applications in random matrix theory. Later chapters use modern convexity theory to establish subtle results about the convergence of eigenvalue distributions as the size of the matrices increases. Random matrices are viewed as geometrical objects with large dimension. The book analyzes the concentration of measure phenomenon, which describes how measures behave on geometrical objects with large dimension. To prove such results for random matrices, the book develops the modern theory of optimal transportation and proves the associated functional inequalities involving entropy and information. These include the logarithmic Sobolev inequality, which measures how fast some physical systems converge to equilibrium.

Author : Roman Vershynin
Publisher : Cambridge University Press
Page : 299 pages
File Size : 50,9 Mb
Release : 2018-09-27
Category : Business & Economics
ISBN : 9781108415194

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High-Dimensional Probability by Roman Vershynin Pdf

An integrated package of powerful probabilistic tools and key applications in modern mathematical data science.

Author : Christian Houdré,David M. Mason,Jan Rosiński,Jon A. Wellner
Publisher : Springer Science & Business Media
Page : 374 pages
File Size : 45,6 Mb
Release : 2013-04-19
Category : Mathematics
ISBN : 9783034804905

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High Dimensional Probability VI by Christian Houdré,David M. Mason,Jan Rosiński,Jon A. Wellner Pdf

This is a collection of papers by participants at High Dimensional Probability VI Meeting held from October 9-14, 2011 at the Banff International Research Station in Banff, Alberta, Canada. High Dimensional Probability (HDP) is an area of mathematics that includes the study of probability distributions and limit theorems in infinite-dimensional spaces such as Hilbert spaces and Banach spaces. The most remarkable feature of this area is that it has resulted in the creation of powerful new tools and perspectives, whose range of application has led to interactions with other areas of mathematics, statistics, and computer science. These include random matrix theory, nonparametric statistics, empirical process theory, statistical learning theory, concentration of measure phenomena, strong and weak approximations, distribution function estimation in high dimensions, combinatorial optimization, and random graph theory. The papers in this volume show that HDP theory continues to develop new tools, methods, techniques and perspectives to analyze the random phenomena. Both researchers and advanced students will find this book of great use for learning about new avenues of research.​

Author : Zhidong Bai,Jack W. Silverstein
Publisher : Springer Science & Business Media
Page : 560 pages
File Size : 41,6 Mb
Release : 2009-12-10
Category : Mathematics
ISBN : 9781441906618

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Spectral Analysis of Large Dimensional Random Matrices by Zhidong Bai,Jack W. Silverstein Pdf

The aim of the book is to introduce basic concepts, main results, and widely applied mathematical tools in the spectral analysis of large dimensional random matrices. The core of the book focuses on results established under moment conditions on random variables using probabilistic methods, and is thus easily applicable to statistics and other areas of science. The book introduces fundamental results, most of them investigated by the authors, such as the semicircular law of Wigner matrices, the Marcenko-Pastur law, the limiting spectral distribution of the multivariate F matrix, limits of extreme eigenvalues, spectrum separation theorems, convergence rates of empirical distributions, central limit theorems of linear spectral statistics, and the partial solution of the famous circular law. While deriving the main results, the book simultaneously emphasizes the ideas and methodologies of the fundamental mathematical tools, among them being: truncation techniques, matrix identities, moment convergence theorems, and the Stieltjes transform. Its treatment is especially fitting to the needs of mathematics and statistics graduate students and beginning researchers, having a basic knowledge of matrix theory and an understanding of probability theory at the graduate level, who desire to learn the concepts and tools in solving problems in this area. It can also serve as a detailed handbook on results of large dimensional random matrices for practical users. This second edition includes two additional chapters, one on the authors' results on the limiting behavior of eigenvectors of sample covariance matrices, another on applications to wireless communications and finance. While attempting to bring this edition up-to-date on recent work, it also provides summaries of other areas which are typically considered part of the general field of random matrix theory.

Author : Martin J. Wainwright
Publisher : Cambridge University Press
Page : 571 pages
File Size : 44,7 Mb
Release : 2019-02-21
Category : Business & Economics
ISBN : 9781108498029

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High-Dimensional Statistics by Martin J. Wainwright Pdf

A coherent introductory text from a groundbreaking researcher, focusing on clarity and motivation to build intuition and understanding.

Author : V.K.B. Kota
Publisher : Springer
Page : 401 pages
File Size : 53,7 Mb
Release : 2014-07-08
Category : Science
ISBN : 9783319045672

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Embedded Random Matrix Ensembles in Quantum Physics by V.K.B. Kota Pdf

Although used with increasing frequency in many branches of physics, random matrix ensembles are not always sufficiently specific to account for important features of the physical system at hand. One refinement which retains the basic stochastic approach but allows for such features consists in the use of embedded ensembles. The present text is an exhaustive introduction to and survey of this important field. Starting with an easy-to-read introduction to general random matrix theory, the text then develops the necessary concepts from the beginning, accompanying the reader to the frontiers of present-day research. With some notable exceptions, to date these ensembles have primarily been applied in nuclear spectroscopy. A characteristic example is the use of a random two-body interaction in the framework of the nuclear shell model. Yet, topics in atomic physics, mesoscopic physics, quantum information science and statistical mechanics of isolated finite quantum systems can also be addressed using these ensembles. This book addresses graduate students and researchers with an interest in applications of random matrix theory to the modeling of more complex physical systems and interactions, with applications such as statistical spectroscopy in mind.

Author : László Erdős,Horng-Tzer Yau
Publisher : American Mathematical Soc.
Page : 226 pages
File Size : 53,7 Mb
Release : 2017-08-30
Category : Random matrices
ISBN : 9781470436483

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A Dynamical Approach to Random Matrix Theory by László Erdős,Horng-Tzer Yau Pdf

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.

Author : Marc Potters,Jean-Philippe Bouchaud
Publisher : Cambridge University Press
Page : 371 pages
File Size : 48,7 Mb
Release : 2020-12-03
Category : Computers
ISBN : 9781108488082

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A First Course in Random Matrix Theory by Marc Potters,Jean-Philippe Bouchaud Pdf

An intuitive, up-to-date introduction to random matrix theory and free calculus, with real world illustrations and Big Data applications.

Author : Giacomo Livan,Marcel Novaes,Pierpaolo Vivo
Publisher : Springer
Page : 124 pages
File Size : 55,7 Mb
Release : 2018-01-16
Category : Science
ISBN : 9783319708850

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Introduction to Random Matrices by Giacomo Livan,Marcel Novaes,Pierpaolo Vivo Pdf

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.

Author : Jinho Baik,Percy Deift,Toufic Suidan
Publisher : American Mathematical Soc.
Page : 461 pages
File Size : 55,8 Mb
Release : 2016-06-22
Category : Combinatorial analysis
ISBN : 9780821848418

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Combinatorics and Random Matrix Theory by Jinho Baik,Percy Deift,Toufic Suidan Pdf

Over the last fifteen years a variety of problems in combinatorics have been solved in terms of random matrix theory. More precisely, the situation is as follows: the problems at hand are probabilistic in nature and, in an appropriate scaling limit, it turns out that certain key quantities associated with these problems behave statistically like the eigenvalues of a (large) random matrix. Said differently, random matrix theory provides a “stochastic special function theory” for a broad and growing class of problems in combinatorics. The goal of this book is to analyze in detail two key examples of this phenomenon, viz., Ulam's problem for increasing subsequences of random permutations and domino tilings of the Aztec diamond. Other examples are also described along the way, but in less detail. Techniques from many different areas in mathematics are needed to analyze these problems. These areas include combinatorics, probability theory, functional analysis, complex analysis, and the theory of integrable systems. The book is self-contained, and along the way we develop enough of the theory we need from each area that a general reader with, say, two or three years experience in graduate school can learn the subject directly from the text.

Author : Elizabeth S. Meckes
Publisher : Cambridge University Press
Page : 225 pages
File Size : 43,7 Mb
Release : 2019-08
Category : Mathematics
ISBN : 9781108419529

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The Random Matrix Theory of the Classical Compact Groups by Elizabeth S. Meckes Pdf

Provides a comprehensive introduction to the theory of random orthogonal, unitary, and symplectic matrices.

Author : Gregory S. Chirikjian
Publisher : Springer Science & Business Media
Page : 461 pages
File Size : 40,8 Mb
Release : 2011-11-16
Category : Mathematics
ISBN : 9780817649449

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Stochastic Models, Information Theory, and Lie Groups, Volume 2 by Gregory S. Chirikjian Pdf

This unique two-volume set presents the subjects of stochastic processes, information theory, and Lie groups in a unified setting, thereby building bridges between fields that are rarely studied by the same people. Unlike the many excellent formal treatments available for each of these subjects individually, the emphasis in both of these volumes is on the use of stochastic, geometric, and group-theoretic concepts in the modeling of physical phenomena. Stochastic Models, Information Theory, and Lie Groups will be of interest to advanced undergraduate and graduate students, researchers, and practitioners working in applied mathematics, the physical sciences, and engineering. Extensive exercises, motivating examples, and real-world applications make the work suitable as a textbook for use in courses that emphasize applied stochastic processes or differential geometry.

Author : Van H. Vu
Publisher : American Mathematical Society
Page : 186 pages
File Size : 52,7 Mb
Release : 2014-07-16
Category : Mathematics
ISBN : 9780821894712

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Modern Aspects of Random Matrix Theory by Van H. Vu Pdf

The theory of random matrices is an amazingly rich topic in mathematics. Random matrices play a fundamental role in various areas such as statistics, mathematical physics, combinatorics, theoretical computer science, number theory and numerical analysis. This volume is based on lectures delivered at the 2013 AMS Short Course on Random Matrices, held January 6-7, 2013 in San Diego, California. Included are surveys by leading researchers in the field, written in introductory style, aiming to provide the reader a quick and intuitive overview of this fascinating and rapidly developing topic. These surveys contain many major recent developments, such as progress on universality conjectures, connections between random matrices and free probability, numerical algebra, combinatorics and high-dimensional geometry, together with several novel methods and a variety of open questions.

Author : Anonim
Publisher : World Scientific
Page : 1001 pages
File Size : 41,7 Mb
Release : 2024-06-30
Category : Electronic
ISBN : 8210379456XXX

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by Anonim Pdf