Random Matrices and Non-Commutative Probability

Random Matrices and Non-Commutative Probability
Title Random Matrices and Non-Commutative Probability PDF eBook
Author Arup Bose
Publisher CRC Press
Pages 420
Release 2021-10-26
Genre Mathematics
ISBN 1000458822

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This is an introductory book on Non-Commutative Probability or Free Probability and Large Dimensional Random Matrices. Basic concepts of free probability are introduced by analogy with classical probability in a lucid and quick manner. It then develops the results on the convergence of large dimensional random matrices, with a special focus on the interesting connections to free probability. The book assumes almost no prerequisite for the most part. However, familiarity with the basic convergence concepts in probability and a bit of mathematical maturity will be helpful. Combinatorial properties of non-crossing partitions, including the Möbius function play a central role in introducing free probability. Free independence is defined via free cumulants in analogy with the way classical independence can be defined via classical cumulants. Free cumulants are introduced through the Möbius function. Free product probability spaces are constructed using free cumulants. Marginal and joint tracial convergence of large dimensional random matrices such as the Wigner, elliptic, sample covariance, cross-covariance, Toeplitz, Circulant and Hankel are discussed. Convergence of the empirical spectral distribution is discussed for symmetric matrices. Asymptotic freeness results for random matrices, including some recent ones, are discussed in detail. These clarify the structure of the limits for joint convergence of random matrices. Asymptotic freeness of independent sample covariance matrices is also demonstrated via embedding into Wigner matrices. Exercises, at advanced undergraduate and graduate level, are provided in each chapter.

Random Matrices and Non-Commutative Probability

Random Matrices and Non-Commutative Probability
Title Random Matrices and Non-Commutative Probability PDF eBook
Author Arup Bose
Publisher CRC Press
Pages 287
Release 2021-10-26
Genre Mathematics
ISBN 1000458814

Download Random Matrices and Non-Commutative Probability Book in PDF, Epub and Kindle

This is an introductory book on Non-Commutative Probability or Free Probability and Large Dimensional Random Matrices. Basic concepts of free probability are introduced by analogy with classical probability in a lucid and quick manner. It then develops the results on the convergence of large dimensional random matrices, with a special focus on the interesting connections to free probability. The book assumes almost no prerequisite for the most part. However, familiarity with the basic convergence concepts in probability and a bit of mathematical maturity will be helpful. Combinatorial properties of non-crossing partitions, including the Möbius function play a central role in introducing free probability. Free independence is defined via free cumulants in analogy with the way classical independence can be defined via classical cumulants. Free cumulants are introduced through the Möbius function. Free product probability spaces are constructed using free cumulants. Marginal and joint tracial convergence of large dimensional random matrices such as the Wigner, elliptic, sample covariance, cross-covariance, Toeplitz, Circulant and Hankel are discussed. Convergence of the empirical spectral distribution is discussed for symmetric matrices. Asymptotic freeness results for random matrices, including some recent ones, are discussed in detail. These clarify the structure of the limits for joint convergence of random matrices. Asymptotic freeness of independent sample covariance matrices is also demonstrated via embedding into Wigner matrices. Exercises, at advanced undergraduate and graduate level, are provided in each chapter.

Free Probability and Random Matrices

Free Probability and Random Matrices
Title Free Probability and Random Matrices PDF eBook
Author James A. Mingo
Publisher Springer
Pages 343
Release 2017-06-24
Genre Mathematics
ISBN 1493969420

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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.

An Introduction to Random Matrices

An Introduction to Random Matrices
Title An Introduction to Random Matrices PDF eBook
Author Greg W. Anderson
Publisher Cambridge University Press
Pages 507
Release 2010
Genre Mathematics
ISBN 0521194520

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

Topics in Random Matrix Theory

Topics in Random Matrix Theory
Title Topics in Random Matrix Theory PDF eBook
Author Terence Tao
Publisher American Mathematical Soc.
Pages 298
Release 2012-03-21
Genre Mathematics
ISBN 0821874306

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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.

Lectures on the Combinatorics of Free Probability

Lectures on the Combinatorics of Free Probability
Title Lectures on the Combinatorics of Free Probability PDF eBook
Author Alexandru Nica
Publisher Cambridge University Press
Pages 430
Release 2006-09-07
Genre Mathematics
ISBN 0521858526

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This 2006 book is a self-contained introduction to free probability theory suitable for an introductory graduate level course.

Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory

Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory
Title Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory PDF eBook
Author Roland Speicher
Publisher American Mathematical Soc.
Pages 105
Release 1998
Genre Mathematics
ISBN 0821806939

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Free probability theory, introduced by Voiculescu, has developed very actively in the last few years and has had an increasing impact on quite different fields in mathematics and physics. Whereas the subject arose out of the field of von Neumann algebras, presented here is a quite different view of Voiculescu's amalgamated free product. This combinatorial description not only allows re-proving of most of Voiculescu's results in a concise and elegant way, but also opens the way for many new results. Unlike other approaches, this book emphasizes the combinatorial structure of the concept of ``freeness''. This gives an elegant and easily accessible description of freeness and leads to new results in unexpected directions. Specifically, a mathematical framework for otherwise quite ad hoc approximations in physics emerges.