Random Matrices and Non-Commutative Probability

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/Non-commutative Probability has gained much attentionsignificant advances have been made since its initiation in the 1990's. Though it started as a branch of Mathematics, it has found significant applications in statistics, wireless communication etc, particularly through deep and interesting connections with Random Matrix Theory. Cover 1 Half Title 2 Title Page 4 Copyright Page 5 Dedication 6 Contents 8 Preface 12 About the Author 14 Notation 16 Introduction 18 1. Classical independence, moments and cumulants 24 1.1. Classical independence 24 1.2. CLT via cumulants 27 1.3. Cumulants to moments 29 1.4. Moments to cumulants, the Möbius function 32 1.5. Classical Isserlis’ formula 34 1.6. Exercises 35 2. Non-commutative probability 38 2.1. Non-crossing partition 38 2.2. Free cumulants 40 2.3. Free Gaussian or semi-circular law 41 2.4. Free Poisson law 44 2.5. Non-commutative and *-probability spaces 45 2.6. Moments and probability laws of variables 48 2.7. Exercises 52 3. Free independence 54 3.1. Free independence 54 3.2. Free product of *-probability spaces 57 3.3. Free binomial 58 3.4. Semi-circular family 58 3.5. Free Isserlis’ formula 59 3.6. Circular and elliptic variables 60 3.7. Free additive convolution 62 3.8. Kreweras complement 63 3.9. Moments of free variables 67 3.10. Compound free Poisson 69 3.11. Exercises 70 4. Convergence 72 4.1. Algebraic convergence 72 4.2. Free central limit theorem 76 4.3. Free Poisson convergence 78 4.4. Sums of triangular arrays 80 4.5. Exercises 81 5. Transforms 84 5.1. Stieltjes transform 84 5.2. R transform 90 5.3. Interrelation 92 5.4. S-transform 97 5.5. Free infinite divisibility 102 5.6. Exercises 108 6. C*-probability space 110 6.1. C*-probability space 110 6.2. Spectrum 111 6.3. Distribution of a self-adjoint element 122 6.4. Free product of C*-probability spaces 125 6.5. Free additive and multiplicative convolution 126 6.6. Exercises 128 7. Random matrices 130 7.1. Empirical spectral measure 130 7.2. Limiting spectral measure 131 7.3. Moment and trace 132 7.4. Some important matrices 133 7.5. A unified treatment 140 7.6. Exercises 144 8. Convergence of some important matrices 146 8.1. Wigner matrix: semi-circular law 146 8.2. S-matrix: Marčenko-Pastur law 150 8.3. IID and elliptic matrices: circular and elliptic variables 158 8.4. Toeplitz matrix 161 8.5. Hankel matrix 163 8.6. Reverse Circulant matrix: symmetrized Rayleigh 164 8.7. Symmetric Circulant: Gaussian law 167 8.8. Almost sure convergence of the ESD 169 8.9. Exercises 171 9. Joint convergence I: single pattern 172 9.1. Unified treatment: extension 172 9.2. Wigner matrices: asymptotic freeness 177 9.3. Elliptic matrices: asymptotic freeness 180 9.4. S-matrices in elliptic models: asymptotic freeness 183 9.5. Symmetric Circulants: asymptotic independence 189 9.6. Reverse Circulants: asymptotic half-independence 190 9.7. Exercises 192 10. Joint convergence II: multiple patterns 194 10.1. Multiple patterns: colors and indices 194 10.2. Joint convergence 197 10.3. Two or more patterns at a time 200 10.4. Sum of independent patterned matrices 207 10.5. Discussion 208 10.6. Exercises 209 11. Asymptotic freeness of random matrices 210 11.1. Elliptic, IID, Wigner and S-matrices 210 11.2. Gaussian elliptic, IID, Wigner and deterministic 211 11.3. General elliptic, IID, Wigner and deterministic matrices 218 11.4. S-matrices and embedding 220 11.5. Cross-covariance matrices 223 11.5.1. Pair-correlated cross-covariance; p/n → y ≠ 0 224 11.5.2. Pair correlated cross-covariance; p/n → 0 231 11.6. Wigner and patterned random matrices 235 11.7. Discussion 245 11.8. Exercises 246 12. Brown measure 248 12.1. Brown measure 248 12.2. Exercises 254 13. Tying three loose ends 256 13.1. Möbius function on NC(n) 256 13.2. Equivalence of two freeness definitions 262 13.3. Free product construction 266 13.4. Exercises 272 Bibliography 274 Index 282 Arup;,Bose;,Commutative;,Matrices;,Non;,Probability;,Random Arup,Bose,Commutative,Matrices,Non,Probability,Random