Lectures on the Combinatorics of Free Probability (London Mathematical Society Lecture Note Series)

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 Book Is The First To Give 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.--pub. Desc. Part I. Basic Concepts: 1. Non-commutative Probability Spaces And Distributions; 2. A Case Study Of Non-normal Distribution; 3. C*-probability Spaces; 4. Non-commutative Joint Distributions; 5. Definition And Basic Properties Of Free Independence; 6. Free Product Of *-probability Spaces; 7. Free Product Of C*-probability Spaces; Part Ii. Cumulants: 8. Motivation: Free Central Limit Theorem; 9. Basic Combinatorics I: Non-crossing Partitions; 10. Basic Combinatorics Ii: M S Inversion; 11. Free Cumulants: Definition And Basic Properties; 12. Sums Of Free Random Variables; 13. More About Limit Theorems And Infinitely Divisible Distributions; 14. Products Of Free Random Variables; 15. R-diagonal Elements; Part Iii. Transforms And Models: 16. The R-transform; 17. The Operation Of Boxed Convolution; 18. More On The 1-dimensional Boxed Convolution; 19. The Free Commutator; 20. R-cyclic Matrices; 21. The Full Fock Space Model For The R-transform; 22. Gaussian Random Matrices; 23. Unitary Random Matrices; Notes And Comments; Bibliography; Index. Alexandru Nica, Roland Speicher. Includes Bibliographical References (p. 405-409) And Index. 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 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. 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