Random Matrix Theory

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Introduction to Random Matrices

Giacomo Livan,Marcel Novaes,Pierpaolo Vivo
Introduction to Random Matrices Book PDF

Author : Giacomo Livan,Marcel Novaes,Pierpaolo Vivo
Publisher : Springer
Page : 124 pages
File Size : 47,9 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.

A First Course in Random Matrix Theory

Marc Potters,Jean-Philippe Bouchaud
A First Course in Random Matrix Theory Book PDF

Author : Marc Potters,Jean-Philippe Bouchaud
Publisher : Cambridge University Press
Page : 371 pages
File Size : 47,9 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.

Topics in Random Matrix Theory

Terence Tao
Topics in Random Matrix Theory Book PDF

Author : Terence Tao
Publisher : American Mathematical Society
Page : 296 pages
File Size : 41,5 Mb
Release : 2023-08-24
Category : Mathematics
ISBN : 9781470474591

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Topics in Random Matrix Theory by Terence Tao Pdf

The field of random matrix theory has seen an explosion of activity in recent years, with connections to many areas of mathematics and physics. However, this makes the current state of the field almost too large to survey in a single book. In this graduate text, we focus on one specific sector of the field, namely the spectral distribution of random Wigner matrix ensembles (such as the Gaussian Unitary Ensemble), as well as iid matrix ensembles. The text is largely self-contained and starts with a review of relevant aspects of probability theory and linear algebra. With over 200 exercises, the book is suitable as an introductory text for beginning graduate students seeking to enter the field.

Combinatorics and Random Matrix Theory

Jinho Baik,Percy Deift,Toufic Suidan
Combinatorics and Random Matrix Theory Book PDF

Author : Jinho Baik,Percy Deift,Toufic Suidan
Publisher : American Mathematical Soc.
Page : 461 pages
File Size : 45,7 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.

A Dynamical Approach to Random Matrix Theory

László Erdős,Horng-Tzer Yau
A Dynamical Approach to Random Matrix Theory Book PDF

Author : László Erdős,Horng-Tzer Yau
Publisher : American Mathematical Soc.
Page : 226 pages
File Size : 52,5 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.

Random Matrix Theory and Wireless Communications

Antonia M. Tulino,Sergio Verdú
Random Matrix Theory and Wireless Communications Book PDF

Author : Antonia M. Tulino,Sergio Verdú
Publisher : Now Publishers Inc
Page : 196 pages
File Size : 52,6 Mb
Release : 2004
Category : Computers
ISBN : 193301900X

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Random Matrix Theory and Wireless Communications by Antonia M. Tulino,Sergio Verdú Pdf

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 Matrix Theory with an External Source

Edouard Brézin,Shinobu Hikami
Random Matrix Theory with an External Source Book PDF

Author : Edouard Brézin,Shinobu Hikami
Publisher : Springer
Page : 138 pages
File Size : 47,7 Mb
Release : 2017-01-11
Category : Science
ISBN : 9789811033162

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Random Matrix Theory with an External Source by Edouard Brézin,Shinobu Hikami Pdf

This is a first book to show that the theory of the Gaussian random matrix is essential to understand the universal correlations with random fluctuations and to demonstrate that it is useful to evaluate topological universal quantities. We consider Gaussian random matrix models in the presence of a deterministic matrix source. In such models the correlation functions are known exactly for an arbitrary source and for any size of the matrices. The freedom given by the external source allows for various tunings to different classes of universality. The main interest is to use this freedom to compute various topological invariants for surfaces such as the intersection numbers for curves drawn on a surface of given genus with marked points, Euler characteristics, and the Gromov–Witten invariants. A remarkable duality for the average of characteristic polynomials is essential for obtaining such topological invariants. The analysis is extended to nonorientable surfaces and to surfaces with boundaries.

An Introduction to Random Matrices

Greg W. Anderson,Alice Guionnet,Ofer Zeitouni
An Introduction to Random Matrices Book PDF

Author : Greg W. Anderson,Alice Guionnet,Ofer Zeitouni
Publisher : Cambridge University Press
Page : 507 pages
File Size : 41,7 Mb
Release : 2010
Category : Mathematics
ISBN : 9780521194525

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An Introduction to Random Matrices by Greg W. Anderson,Alice Guionnet,Ofer Zeitouni Pdf

A rigorous introduction to the basic theory of random matrices designed for graduate students with a background in probability theory.

The Random Matrix Theory of the Classical Compact Groups

Elizabeth S. Meckes
The Random Matrix Theory of the Classical Compact Groups Book PDF

Author : Elizabeth S. Meckes
Publisher : Cambridge University Press
Page : 225 pages
File Size : 51,9 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.

The Oxford Handbook of Random Matrix Theory

Gernot Akemann,Jinho Baik,Philippe Di Francesco
The Oxford Handbook of Random Matrix Theory Book PDF

Author : Gernot Akemann,Jinho Baik,Philippe Di Francesco
Publisher : Oxford Handbooks
Page : 0 pages
File Size : 42,7 Mb
Release : 2015-08-09
Category : Mathematics
ISBN : 0198744196

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The Oxford Handbook of Random Matrix Theory by Gernot Akemann,Jinho Baik,Philippe Di Francesco Pdf

With a foreword by Freeman Dyson, the handbook brings together leading mathematicians and physicists to offer a comprehensive overview of random matrix theory, including a guide to new developments and the diverse range of applications of this approach.In part one, all modern and classical techniques of solving random matrix models are explored, including orthogonal polynomials, exact replicas or supersymmetry.

Random Matrix Theory

Percy Deift,Dimitri Gioev
Random Matrix Theory Book PDF

Author : Percy Deift,Dimitri Gioev
Publisher : American Mathematical Soc.
Page : 236 pages
File Size : 49,9 Mb
Release : 2009-01-01
Category : Mathematics
ISBN : 9780821883570

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Random Matrix Theory by Percy Deift,Dimitri Gioev Pdf

"This book features a unified derivation of the mathematical theory of the three classical types of invariant random matrix ensembles-orthogonal, unitary, and symplectic. The authors follow the approach of Tracy and Widom, but the exposition here contains a substantial amount of additional material, in particular, facts from functional analysis and the theory of Pfaffians. The main result in the book is a proof of universality for orthogonal and symplectic ensembles corresponding to generalized Gaussian type weights following the authors' prior work. New, quantitative error estimates are derived." --Book Jacket.

Spectral Analysis of Large Dimensional Random Matrices

Zhidong Bai,Jack W. Silverstein
Spectral Analysis of Large Dimensional Random Matrices Book PDF

Author : Zhidong Bai,Jack W. Silverstein
Publisher : Springer Science & Business Media
Page : 560 pages
File Size : 53,8 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.

Recent Perspectives in Random Matrix Theory and Number Theory

F. Mezzadri,N. C. Snaith
Recent Perspectives in Random Matrix Theory and Number Theory Book PDF

Author : F. Mezzadri,N. C. Snaith
Publisher : Cambridge University Press
Page : 530 pages
File Size : 44,5 Mb
Release : 2005-06-21
Category : Mathematics
ISBN : 9780521620581

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Recent Perspectives in Random Matrix Theory and Number Theory by F. Mezzadri,N. C. Snaith Pdf

Provides a grounding in random matrix techniques applied to analytic number theory.

Free Probability and Random Matrices

James A. Mingo,Roland Speicher
Free Probability and Random Matrices Book PDF

Author : James A. Mingo,Roland Speicher
Publisher : Springer
Page : 336 pages
File Size : 52,7 Mb
Release : 2017-06-24
Category : Mathematics
ISBN : 9781493969425

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Free Probability and Random Matrices by James A. Mingo,Roland Speicher Pdf

This volume opens the world of free probability to a wide variety of readers. From its roots in the theory of operator algebras, free probability has intertwined with non-crossing partitions, random matrices, applications in wireless communications, representation theory of large groups, quantum groups, the invariant subspace problem, large deviations, subfactors, and beyond. This book puts a special emphasis on the relation of free probability to random matrices, but also touches upon the operator algebraic, combinatorial, and analytic aspects of the theory. The book serves as a combination textbook/research monograph, with self-contained chapters, exercises scattered throughout the text, and coverage of important ongoing progress of the theory. It will appeal to graduate students and all mathematicians interested in random matrices and free probability from the point of view of operator algebras, combinatorics, analytic functions, or applications in engineering and statistical physics.

Log-Gases and Random Matrices (LMS-34)

Peter J. Forrester
Log Gases and Random Matrices LMS 34 Book PDF

Author : Peter J. Forrester
Publisher : Princeton University Press
Page : 808 pages
File Size : 46,6 Mb
Release : 2010-07-01
Category : Mathematics
ISBN : 9781400835416

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Log-Gases and Random Matrices (LMS-34) by Peter J. Forrester Pdf

Random matrix theory, both as an application and as a theory, has evolved rapidly over the past fifteen years. Log-Gases and Random Matrices gives a comprehensive account of these developments, emphasizing log-gases as a physical picture and heuristic, as well as covering topics such as beta ensembles and Jack polynomials. Peter Forrester presents an encyclopedic development of log-gases and random matrices viewed as examples of integrable or exactly solvable systems. Forrester develops not only the application and theory of Gaussian and circular ensembles of classical random matrix theory, but also of the Laguerre and Jacobi ensembles, and their beta extensions. Prominence is given to the computation of a multitude of Jacobians; determinantal point processes and orthogonal polynomials of one variable; the Selberg integral, Jack polynomials, and generalized hypergeometric functions; Painlevé transcendents; macroscopic electrostatistics and asymptotic formulas; nonintersecting paths and models in statistical mechanics; and applications of random matrix theory. This is the first textbook development of both nonsymmetric and symmetric Jack polynomial theory, as well as the connection between Selberg integral theory and beta ensembles. The author provides hundreds of guided exercises and linked topics, making Log-Gases and Random Matrices an indispensable reference work, as well as a learning resource for all students and researchers in the field.