Random Matrices and Non-Commutative Probability  book cover
1st Edition

Random Matrices and Non-Commutative Probability

ISBN 9780367700812
Published October 27, 2021 by Chapman and Hall/CRC
286 Pages

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Book Description

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.

Table of Contents


  1. Classical independence, moments and cumulants
  2. Classical independence

    CLT a cumulants

    Cumulants to moments

    Moments to cumulants, the Möbius function

    Classical Isserlis’ formula


  3. Non-commutative probability
  4. Non-crossing partition

    Free cumulants

    Free Gaussian or semi-circle law

    Free Poisson law

    Non-commutative and *-probability spaces

    Moments and probability laws of variables


  5. Free independence
  6. Free independence

    Free product of *-probability spaces

    Free binomial

    Semi-circular family

    Free Isserlis’ formula

    Circular and elliptic variables

    Free additive convolution

    Kreweras complement

    Moments of free variables

    0 Compound free Poisson


  7. Convergence
  8. Algebraic convergence

    Free central limit theorem

    Free Poisson convergence

    Sums of triangular arrays


  9. Transforms
  10. Stieltjes transform

    R transform



    Free infinite divisibility


  11. C* -probability space
  12. C* -probability space


    Distribution of a self-adjoint element

    Free product of C* -probability spaces

    Free additive and multiplicative convolution


  13. Random matrices
  14. Empirical spectral measure

    Limiting spectral measure

    Moment and trace

    Some important matrices

    A unified treatment


  15. Convergence of some important matrices
  16. Wigner matrix: semi-circle law

    S-matrix: Marcenko-Pastur law

    IID and elliptic matrices: circular and elliptic variables

    Toeplitz matrix

    Hankel matrix

    Reverse Circulant matrix: symmetrized Rayleigh law

    Symmetric Circulant: Gaussian law

    Almost sure convergence of the ESD


  17. Joint convergence I: single pattern
  18. Unified treatment: extension

    Wigner matrices: asymptotic freeness

    Elliptic matrices: asymptotic freeness

    S-matrices in elliptic models: asymptotic freeness

    Symmetric Circulants: asymptotic independence

    Reverse Circulants: asymptotic half independence


  19. Joint convergence II: multiple patterns
  20. Multiple patterns: colors and indices

    Joint convergence

    Two or more patterns at a time

    Sum of independent patterned matrices



  21. Asymptotic freeness of random matrices
  22. Elliptic, IID, Wigner, and S-matrices

    Gaussian elliptic, IID, Wigner and deterministic matrices

    General elliptic, IID, Wigner, and deterministic matrices

    S-matrices and embedding

    Cross covariance matrices

    Pair-correlated cross-covariance; p/n ! y

    Pair correlated cross covariance; p/n !

    Wigner and patterned random matrices



  23. Brown measure
  24. Brown measure


  25. Tying three loose ends

Möbius function on NC(n)

Equivalence of two freeness definitions

Free product construction




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Arup Bose is on the faculty of the Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Kolkata, India. He has research contributions in statistics, probability, economics and econometrics. He is a Fellow of the Institute of Mathematical Statistics (USA), and of all three national science academies of India. He is a recipient of the S.S. Bhatnagar Prize and the C.R. Rao Award and holds a J.C.Bose National Fellowship. He has been on the editorial board of several journals. He has authored four books: Patterned Random Matrices, Large Covariance and Autocovariance Matrices (with Monika Bhattacharjee), U-Statistics, Mm-Estimators and Resampling (with Snigdhansu Chatterjee) and Random Circulant Matrices (with Koushik Saha).