**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** |