Random Matrices and Non-Commutative Probability

This book provides an introduction to Non-Commutative Probability or Free Probability and Large Dimensional Random Matrices. It assumes almost no prerequisite and covers topics such as free independence, free cumulants, free product probability spaces, marginal and joint tracial convergence, empirical spectral distribution, asymptotic freeness, and exercises.

Format: Paperback / softback
Length: 264 pages
Publication date: 29 January 2024
Publisher: Taylor & Francis Ltd

This comprehensive textbook delves into the realm of Non-Commutative Probability, also known as Free Probability, and explores the fascinating world of Large Dimensional Random Matrices. It serves as an introductory guide, presenting fundamental concepts of free probability in a clear and concise manner, drawing parallels with classical probability for a seamless understanding. The book then proceeds to delve into the convergence of large dimensional random matrices, with a special emphasis on the intriguing connections to free probability. While the book assumes minimal prerequisites, a basic understanding of probability convergence concepts and a certain level of mathematical maturity will undoubtedly enhance the reader's journey.

Combinatorial properties of non-crossing partitions, such as the Möbius function, take center stage in introducing free probability. Free independence is defined through free cumulants, akin to classical independence's definition through classical cumulants. These cumulants are introduced through the Möbius function, laying the foundation for the construction of free product probability spaces.

The book discusses the marginal and joint tracial convergence of various large dimensional random matrices, including the Wigner, elliptic, sample covariance, cross-covariance, Toeplitz, Circulant, and Hankel matrices. It explores the convergence of the empirical spectral distribution for symmetric matrices and delves into detailed asymptotic freeness results for random matrices. These results clarify the structure of joint convergence limits and demonstrate the asymptotic freeness of independent sample covariance matrices through embedding into Wigner matrices.

Each chapter concludes with a comprehensive set of exercises, designed to challenge advanced undergraduate and graduate students. This textbook serves as an invaluable resource for students and researchers interested in Non-Commutative Probability, Free Probability, and Large Dimensional Random Matrices, providing a solid foundation for further exploration in these dynamic fields.

Weight: 526g
Dimension: 234 x 156 (mm)
ISBN-13: 9780367705008