Free Probability and Random Matrices
Free Probability and Random Matrices

Author: James A. Mingo

Publisher: Springer

Published: 2017-06-24

Total Pages: 343

ISBN-13: 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.

Free Probability Theory
Free Probability Theory

Author: Dan V. Voiculescu

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 322

ISBN-13: 0821806750

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This is a volume of papers from a workshop on Random Matrices and Operator Algebra Free Products, held at The Fields Institute for Research in the Mathematical Sciences in March 1995. Over the last few years, there has been much progress on the operator algebra and noncommutative probability sides of the subject. New links with the physics of masterfields and the combinatorics of noncrossing partitions have emerged. Moreover there is a growing free entropy theory.

Introduction to Probability
Introduction to Probability

Author: David F. Anderson

Publisher: Cambridge University Press

Published: 2017-11-02

Total Pages: 447

ISBN-13: 110824498X

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This classroom-tested textbook is an introduction to probability theory, with the right balance between mathematical precision, probabilistic intuition, and concrete applications. Introduction to Probability covers the material precisely, while avoiding excessive technical details. After introducing the basic vocabulary of randomness, including events, probabilities, and random variables, the text offers the reader a first glimpse of the major theorems of the subject: the law of large numbers and the central limit theorem. The important probability distributions are introduced organically as they arise from applications. The discrete and continuous sides of probability are treated together to emphasize their similarities. Intended for students with a calculus background, the text teaches not only the nuts and bolts of probability theory and how to solve specific problems, but also why the methods of solution work.

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

Author: Roland Speicher

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 105

ISBN-13: 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.

Free Random Variables
Free Random Variables

Author: Dan V. Voiculescu

Publisher: American Mathematical Soc.

Published: 1992

Total Pages: 80

ISBN-13: 0821811401

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This book presents the first comprehensive introduction to free probability theory, a highly noncommutative probability theory with independence based on free products instead of tensor products. Basic examples of this kind of theory are provided by convolution operators on free groups and by the asymptotic behavior of large Gaussian random matrices. The probabilistic approach to free products has led to a recent surge of new results on the von Neumann algebras of free groups. The book is ideally suited as a textbook for an advanced graduate course and could also provide material for a seminar. In addition to researchers and graduate students in mathematics, this book will be of interest to physicists and others who use random matrices.

Free Probability and Operator Algebras
Free Probability and Operator Algebras

Author: Dan V. Voiculescu

Publisher: European Mathematical Society

Published: 2016

Total Pages: 148

ISBN-13: 9783037191651

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Free probability is a probability theory dealing with variables having the highest degree of noncommutativity, an aspect found in many areas (quantum mechanics, free group algebras, random matrices, etc.). Thirty years after its foundation, it is a well-established and very active field of mathematics. Originating from Voiculescu's attempt to solve the free group factor problem in operator algebras, free probability has important connections with random matrix theory, combinatorics, harmonic analysis, representation theory of large groups, and wireless communication. These lecture notes arose from a master class in Munster, Germany and present the state of free probability from an operator algebraic perspective. This volume includes introductory lectures on random matrices and combinatorics of free probability (Speicher), free monotone transport (Shlyakhtenko), free group factors (Dykema), free convolution (Bercovici), easy quantum groups (Weber), and a historical review with an outlook (Voiculescu). To make it more accessible, the exposition features a chapter on the basics of free probability and exercises for each part. This book is aimed at master students to early career researchers familiar with basic notions and concepts from operator algebras.

Probability
Probability

Author: Rick Durrett

Publisher: Cambridge University Press

Published: 2010-08-30

Total Pages:

ISBN-13: 113949113X

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This classic introduction to probability theory for beginning graduate students covers laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems. The fourth edition begins with a short chapter on measure theory to orient readers new to the subject.

Basic Probability Theory
Basic Probability Theory

Author: Robert B. Ash

Publisher: Courier Corporation

Published: 2008-06-26

Total Pages: 354

ISBN-13: 0486466280

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This introduction to more advanced courses in probability and real analysis emphasizes the probabilistic way of thinking, rather than measure-theoretic concepts. Geared toward advanced undergraduates and graduate students, its sole prerequisite is calculus. Taking statistics as its major field of application, the text opens with a review of basic concepts, advancing to surveys of random variables, the properties of expectation, conditional probability and expectation, and characteristic functions. Subsequent topics include infinite sequences of random variables, Markov chains, and an introduction to statistics. Complete solutions to some of the problems appear at the end of the book.