Lectures on the combinatorics of free probability

Free Probability Theory studies a special class of 'noncommutative'random variables, which appear in the context of operators on Hilbert spaces and in one of the large random matrices. Since its emergence in the 1980s, free probability has evolved into an established field of mathematics with strong connections to other mathematical areas, such as operator algebras, classical probability theory, random matrices, combinatorics, representation theory of symmetric groups. Free probability also connects to more applied scientific fields, such as wireless communication in electrical engineering. This 2006 book gives a self-contained and comprehensive introduction to free probability theory which has its main focus on the combinatorial aspects. The volume is designed so that it can be used as a text for an introductory course (on an advanced undergraduate or beginning graduate level), and is also well-suited for the individual study of free probability.

Introduction	p. xiii
Part 1 Basic concepts	p. 1
Lecture 1 Non-commutative probability spaces and distributions	p. 3
Non-commutative probability spaces	p. 3
*-distributions (case of normal elements)	p. 7
*-distributions (general case)	p. 13
Exercises	p. 15
Lecture 2 A case study of non-normal distribution	p. 19
Description of the example	p. 19
Dyck paths	p. 22
The distribution of a + a[superscript *]	p. 26
Using the Cauchy transform	p. 30
Exercises	p. 33
Lecture 3 C[superscript *]-probability spaces	p. 35
Functional calculus in a C[superscript *]-algebra	p. 35
C[superscript *]-probability spaces	p. 39
*-distribution, norm and spectrum for a normal element	p. 43
Exercises	p. 46
Lecture 4 Non-commutative joint distributions	p. 49
Joint distributions	p. 49
Joint *-distributions	p. 53
Joint *-distributions and isomorphism	p. 55
Exercises	p. 59
Lecture 5 Definition and basic properties of free independence	p. 63
The classical situation: tensor independence	p. 63
Definition of free independence	p. 64
The example of a free product of groups	p. 66
Free independence and joint moments	p. 69
Some basic properties of free independence	p. 71
Are there other universal product constructions?	p. 75
Exercises	p. 78
Lecture 6 Free product of *-probability spaces	p. 81
Free product of unital algebras	p. 81
Free product of non-commutative probability spaces	p. 84
Free product of *-probability spaces	p. 86
Exercises	p. 92
Lecture 7 Free product of C[superscript *]-probability spaces	p. 95
The GNS representation	p. 95
Free product of C[superscript *]-probability spaces	p. 99
Example: semicircular systems and the full Fock space	p. 102
Exercises	p. 109
Part 2 Cumulants	p. 113
Lecture 8 Motivation: free central limit theorem	p. 115
Convergence in distribution	p. 115
General central limit theorem	p. 117
Classical central limit theorem	p. 120
Free central limit theorem	p. 121
The multi-dimensional case	p. 125
Conclusion and outlook	p. 131
Exercises	p. 132
Lecture 9 Basic combinatorics I: non-crossing partitions	p. 135
Non-crossing partitions of an ordered set	p. 135
The lattice structure of NC(n)	p. 144
The factorization of intervals in NC	p. 148
Exercises	p. 153
Lecture 10 Basic combinatorics II: Mobius inversion	p. 155
Convolution in the framework of a poset	p. 155
Mobius inversion in a lattice	p. 160
The Mobius function of NC	p. 162
Multiplicative functions on NC	p. 164
Functional equation for convolution with [Mu subscript n]	p. 168
Exercises	p. 171
Lecture 11 Free cumulants: definition and basic properties	p. 173
Multiplicative functionals on NC	p. 173
Definition of free cumulants	p. 175
Products as arguments	p. 178
Free independence and free cumulants	p. 182
Cumulants of random variables	p. 185
Example: semicircular and circular elements	p. 187
Even elements	p. 188
Appendix Classical cumulants	p. 190
Exercises	p. 193
Lecture 12 Sums of free random variables	p. 195
Free convolution	p. 195
Analytic calculation of free convolution	p. 200
Proof of the free central limit theorem via R-transform	p. 202
Free Poisson distribution	p. 203
Compound free Poisson distribution	p. 206
Exercises	p. 208
Lecture 13 More about limit theorems and infinitely divisible distributions	p. 211
Limit theorem for triangular arrays	p. 211
Cumulants of operators on Fock space	p. 214
Infinitely divisible distributions	p. 215
Conditionally positive definite sequences	p. 216
Characterization of infinitely divisible distributions	p. 220
Exercises	p. 221
Lecture 14 Products of free random variables	p. 223
Multiplicative free convolution	p. 223
Combinatorial description of free multiplication	p. 225
Compression by a free projection	p. 228
Convolution semigroups ([Mu superscript Characters not reproduciblet])[subscript tge]1	p. 231
Compression by a free family of matrix units	p. 233
Exercises	p. 236
Lecture 15 R-diagonal elements	p. 237
Motivation: cumulants of Haar unitary elements	p. 237
Definition of R-diagonal elements	p. 240
Special realizations of tracial R-diagonal elements	p. 245
Product of two free even elements	p. 249
The free anti-commutator of even elements	p. 251
Powers of R-diagonal elements	p. 253
Exercises	p. 254
Part 3 Transforms and models	p. 257
Lecture 16 The R-transform	p. 259
The multi-variable R-transform	p. 259
The functional equation for the R-transform	p. 265
More about the one-dimensional case	p. 269
Exercises	p. 272
Lecture 17 The operation of boxed convolution	p. 273
The definition of boxed convolution, and its motivation	p. 273
Basic properties of boxed convolution	p. 275
Radial series	p. 277
The Mobius series and its use	p. 280
Exercises	p. 285
Lecture 18 More on the one-dimensional boxed convolution	p. 287
Relation to multiplicative functions on NC	p. 287
The S-transform	p. 293
Exercises	p. 300
Lecture 19 The free commutator	p. 303
Free commutators of even elements	p. 303
Free commutators in the general case	p. 310
The cancelation phenomenon	p. 314
Exercises	p. 317
Lecture 20 R-cyclic matrices	p. 321
Definition and examples of R-cyclic matrices	p. 321
The convolution formula for an R-cyclic matrix	p. 324
R-cyclic families of matrices	p. 329
Applications of the convolution formula	p. 331
Exercises	p. 335
Lecture 21 The full Fock space model for the R-transform	p. 339
Description of the Fock space model	p. 339
An application: revisiting free compressions	p. 346
Exercises	p. 356
Lecture 22 Gaussian random matrices	p. 359
Moments of Gaussian random variables	p. 359
Random matrices in general	p. 361
Selfadjoint Gaussian random matrices and genus expansion	p. 363
Asymptotic free independence for several independent Gaussian random matrices	p. 368
Asymptotic free independence between Gaussian random matrices and constant matrices	p. 371
Lecture 23 Unitary random matrices	p. 379
Haar unitary random matrices	p. 379
The length function on permutations	p. 381
Asymptotic freeness for Haar unitary random matrices	p. 384
Asymptotic freeness between randomly rotated constant matrices	p. 385
Embedding of non-crossing partitions into permutations	p. 390
Exercises	p. 393
Notes and comments	p. 395
References	p. 405
Index	p. 411

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