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Summary
Summary
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.
Table of Contents
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 |