Lectures on the combinatorics of free probability

Cover......Page 1
London Mathematical Society Lecture Note Series 335......Page 2
Lectures on the Combinatorics of Free Probability......Page 4
0521858526......Page 5
Contents......Page 8
Introduction......Page 14
Part 1. Basic concepts......Page 18
Non-commutative probability spaces......Page 20
*-distributions (case of normal elements)......Page 24
*-distributions (general case)......Page 30
Exercises......Page 32
Description of the example......Page 36
Dyck paths......Page 39
The distribution of a + a*......Page 43
Using the Cauchy transform......Page 47
Exercises......Page 50
Functional calculus in a C*-algebra......Page 52
C*-probability spaces......Page 56
*-distribution, norm and spectrum for a normal element......Page 60
Exercises......Page 63
Joint distributions......Page 66
Joint *-distributions......Page 70
Joint *-distributions and isomorphism......Page 72
Exercises......Page 76
The classical situation: tensor independence......Page 80
Definition of free independence......Page 81
The example of a free product of groups......Page 83
Free independence and joint moments......Page 86
Some basic properties of free independence......Page 88
Are there other universal product constructions?......Page 92
Exercises......Page 95
Free product of unital algebras......Page 98
Free product of non-commutative probability spaces......Page 101
Free product of *-probability spaces......Page 103
Exercises......Page 109
The GNS representation......Page 112
Free product of C*-probability spaces......Page 116
Example: semicircular systems and the full Fock space......Page 119
Exercises......Page 126
Part 2. Cumulants......Page 130
Convergence in distribution......Page 132
General central limit theorem......Page 134
Classical central limit theorem......Page 137
Free central limit theorem......Page 138
The multi-dimensional case......Page 142
Conclusion and outlook......Page 148
Exercises......Page 149
Non-crossing partitions of an ordered set......Page 152
The lattice structure of NC(n)......Page 161
The factorization of intervals in NC......Page 165
Exercises......Page 170
Convolution in the framework of a poset......Page 172
Mobius inversion in a lattice......Page 177
The Mobius function of NC......Page 179
Multiplicative functions on NC......Page 181
Functional equation for convolution with µ_n......Page 185
Exercises......Page 188
Multiplicative functionals on NC......Page 190
Definition of free cumulants......Page 192
Products as arguments......Page 195
Free independence and free cumulants......Page 199
Cumulants of random variables......Page 202
Example: semicircular and circular elements......Page 204
Even elements......Page 205
Appendix: classical cumulants......Page 207
Exercises......Page 210
Free convolution......Page 212
Analytic calculation of free convolution......Page 217
Proof of the free central limit theorem via R-transform......Page 219
Free Poisson distribution......Page 220
Compound free Poisson distribution......Page 223
Exercises......Page 225
Limit theorem for triangular arrays......Page 228
Cumulants of operators on Fock space......Page 231
Infinitely divisible distributions......Page 232
Conditionally positive definite sequences......Page 233
Characterization of infinitely divisible distributions......Page 237
Exercises......Page 238
Multiplicative free convolution......Page 240
Combinatorial description of free multiplication......Page 242
Compression by a free projection......Page 245
Convolution semigroups (μ^{\\boxplus t})_{t≥1}......Page 248
Compression by a free family of matrix units......Page 250
Exercises......Page 253
Motivation: cumulants of Haar unitary elements......Page 254
Definition of R-diagonal elements......Page 257
Special realizations of tracial R-diagonal elements......Page 262
Product of two free even elements......Page 266
The free anti-commutator of even elements......Page 268
Powers of R-diagonal elements......Page 270
Exercises......Page 271
Part 3. Transforms and models......Page 274
The multi-variable R-transform......Page 276
The functional equation for the R-transform......Page 282
More about the one-dimensional case......Page 286
Exercises......Page 289
The definition of boxed convolution, and its motivation......Page 290
Basic properties of boxed convolution......Page 292
Radial series......Page 294
The M¨obius series and its use......Page 297
Exercises......Page 302
Relation to multiplicative functions on NC......Page 304
The S-transform......Page 310
Exercises......Page 317
Free commutators of even elements......Page 320
Free commutators in the general case......Page 327
The cancelation phenomenon......Page 331
Exercises......Page 334
Definition and examples of R-cyclic matrices......Page 338
The convolution formula for an R-cyclic matrix......Page 341
R-cyclic families of matrices......Page 346
Applications of the convolution formula......Page 348
Exercises......Page 352
Description of the Fock space model......Page 356
An application: revisiting free compressions......Page 363
Exercises......Page 373
Moments of Gaussian random variables......Page 376
Random matrices in general......Page 378
Selfadjoint Gaussian random matrices and genus expansion......Page 380
Asymptotic free independence for several independent Gaussian random matrices......Page 385
Asymptotic free independence between Gaussian random matrices and constant matrices......Page 388
Haar unitary random matrices......Page 396
The length function on permutations......Page 398
Asymptotic freeness for Haar unitary random matrices......Page 401
Asymptotic freeness between randomly rotated constant matrices......Page 402
Embedding of non-crossing partitions into permutations......Page 407
Exercises......Page 410
Notes and comments......Page 412
References......Page 422
Index......Page 428