1st Edition

Random Matrices and Non-Commutative Probability

By Arup Bose Copyright 2022
    286 Pages
    by Chapman & Hall

    286 Pages
    by Chapman & Hall

    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.


    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





    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).