| Preface | 3 |
| Overview | 5 |
| 1 Probability Laws and Noncommutative Random Variables | 23 |
| 1.1 Distribution measure of normal operators | 24 |
| 1.2 Noncommutative random variables | 35 |
| 2. The Free Relation | 43 |
| 2.1 The free product | 44 |
| 2.2 The free relation | 46 |
| 2.3 The free central limit theorem | 52 |
| 2.4 Free convolution of measures | 56 |
| 2.5 Moments and cumulants | 64 |
| 2.6 Multivariables | 75 |
| 3. Analytic Function Theory and Infinitely Divisible Laws | 95 |
| 3.1 Cauchy transform, Poisson integral, and Hilbert transform | 96 |
| 3.2 Relation between Cauchy transform and R-series | 99 |
| 3.3 Infinitely divisible laws | 102 |
| 4. Random Matrices and Asymptotically Free Relation | 117 |
| 4.1 Random matrices and their eigenvalues | 118 |
| 4.2 Random unitary matrices and asymptotic freeness | 139 |
| 4.3 Asymptotic freeness of some random matirces | 150 |
| 4.4 Random matrix models of noncommutative random variables | 165 |
| 5. Large Deviations for Random Matrices | 179 |
| 5.1 Boltzmann entropy and large deviations | 1801 |
| 5.2 Entropy and random matrices | 185 |
| 5.3 Logarithmic energy and free entropy | 193 |
| 5.4 Gaussian and unitary random matrices | 213 |
| 5.5 The Wishart matrix | 229 |
| 5.6 Entropy and large diviations revisited | 242 |
| 6. Free Entropy of Noncommutative Random Variables | 249 |
| 6.1 Definition and basic properties | 250 |
| 6.2 Calculus for power series of noncommutative variables | 257 |
| 6.3 Change of variable formulas for free entropy | 263 |
| 6.4 Additivity of free entropy | 273 |
| 6.5 Free entropies of unitary and non-selfadjoint random variables | 279 |
| 6.6 Relation between different free entropies | 284 |
| 7 Relation to Operator Algebras | 305 |
| 7.1 Free Group factors and semicircular systems | 306 |
| 7.2 Interpolar free group factors | 314 |
| 7.3 Free entropy dimension | 328 |
| 7.4 Applications of free entropy | 338 |
| Bibliography | 347 |
| Index | 359 |