Free probability and random matrices:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
[2017]
|
| Series: | Fields Institute monographs
Volume 35 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | xiv, 336 Seiten Diagramme 23.5 cm x 15.5 cm |
| ISBN: | 9781493969418 1493969412 |
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Record in the Search Index
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Contents 1 Asymptotic Freeness of Gaussian Random Matrices. 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1,12 1.13 1.14 2 The Free Central Limit Theorem and Free Cumulants. . 2.1 2.2 2.3 2.4 2.5 3 Moments and cumulants of random variables. Moments of a Gaussian random variable. Gaussian vectors. The moments of a standard complex Gaussian random variable. Wick’s formula. Gaussian random matrices. A genus expansion for the GUE. Non-crossing partitions and permutations. Wigner’s semi-circle law. Asymptotic freeness of independent GUE’s. Freeness and asymptotic freeness. Basic properties of freeness. Classical moment-cumulant formulas . Additional exercises . The classical and free central limit theorems. 2.1.1 Classical
central limit theorem. 2.1.2 Free central limit theorem . Non-crossing partitions and free cumulants. Products of free random variables. Functional relation between moment series and cumulant series. Subordination and the non-commutative derivative. Free Harmonic Analysis. 3.1 3.2 3.3 The Cauchy transform. Moments and asymptotic expansions. Analyticity of the R-transform: compactly supported measures. 67 1 2 3 5 5 6 7 8 9 11 12 14 15 18 20 23 23 28 28 32 39 41 44 51 52 66 vii
viii Contents 3.4 3.5 3.6 4 Measures with finite variance. The free additive convolution of probability measures with finite variance. 77 The R-transform and free additive convolution of arbitrary measures. 82 Asymptotic Freeness for Gaussian, Wigner, and Unitary Random Matrices. 93 4.1 Asymptotic freeness: averaged convergence versus almost sure convergence. 93 4.2 Asymptotic freeness of Gaussian random matrices and deterministic matrices. 99 4.3 Asymptotic freeness of Haar distributed unitary random matrices and deterministic matrices. 102 4.4 Asymptotic freeness between Wigner and deterministic random matrices . 106 4.5 Examples of random matrix calculations. 4.5.1 Wishart matrices and the Marchenko-Pastur distribution. 4.5.2 Sum of random matrices. 4.5.3 Product of random matrices . 71 116 116 118 119 5 Fluctuations and Second Order Freeness.
121 5.1 Fluctuations of GUE random matrices. 122 5.2 Fluctuations of several matrices. 132 5.3 Second order probability space and second order freeness. 136 5.4 Second order cumulants. 142 5.5 Functional relation between second order moment and cumulant series. 149 5.6 Diagonalization of fluctuations. 151 5.6.1 Diagonalization in the one-matrix case. 152 5.6.2 Diagonalization in the multivariate case. 158 6 Free Group Factors and Freeness . 6.1 Group (von Neumann) algebras. 6.2 Free group factors. 6.3 Free product of groups. 6.4 Moments and isomorphism of von Neumann algebras . 6.5 Freeness in the free group factors. 6.6 The structure of free group factors. 6.7 Compression of free group factors. 6.8 Circular operators and complex Gaussian randommatrices. 6.9 Proof of C(F3)1/2 .
. 6.10 The general case C(Fn)1/k C(F,+(n-Y)k2) . 6.11 Interpolating free group factors. 6.12 The dichotomy for the free group factor isomorphism problem. 159 160 161 162 162 163 165 166 168 171 173 173 174
Contents ix 7 Free Entropy Z : The Microstates Approach via Large Deviations. 175 7.1 Motivation. 175 7.2 Large deviation theory and Cramér’s theorem. 176 7.3 S anov ’ s theorem and entropy. 181 7.4 Back to random matrices and one-dimensional free entropy. 182 7.5 Definition of multivariate free entropy. 185 7.6 Some important properties of X. 187 7.7 Applications of free entropy to operator algebras. 189 7.7.1 The proof of Theorem 7, part (i ). 191 7.7.2 The proof of Theorem 7, part (Ui). 193 8 Free Entropy X*: The Non-microstates Approach via Free Fisher Information. 195 8.1 Non-commutative derivatives. 8.2 9j as unbounded operator on C(xi,., xn). 8.3 Conjugate variables and free Fisher information Φ*. 8.4 Additivity of Φ* and freeness. 8.5 The non-microstates free entropy X* . 8.6 Operator algebraic applications of free Fisher information. 8.7 Absence of atoms for self-adjoint
polynomials. 8.8 Additional exercises. 196 198 203 212 216 217 219 221 Operator-Valued Free Probability Theory and Block Random Matrices. 9.1 Gaussian block random matrices. 9.2 General theory of operator-valued free probability . 9.3 Relation between scalar-valued and matrix-valued cumulants . 9.4 Moving between different levels. 9.5 A non-self-adjoint example. . 225 225 234 240 242 245 9 10 11 Deterministic Equivalents, Polynomials in Free Variables, and Analytic Theory of Operator-Valued Convolution. 249 10.1 The general concept of a free deterministic equivalent. 10.2 A motivating example: reduction to multiplicative convolution. 10.3 The general case: reduction to operator-valued additive convolution via the linearization trick. 253 10.4 Analytic theory of operator-valued convolutions. 10.4.1 General notations. 10.4.2 Operator-valued additive convolution. 10.4.3 Operator-valued multiplicative convolution. 10.5 Numerical
example. 10.6 The case of rational functions. 10.7 Additional exercise . 249 252 257 258 258 259 259 260 262 Brown Measure. 263 11.1 Brown measure for normal operators. 263 11.2 Brown measure for matrices. 264
Contents X Fuglede-Kadison determinant in finite von Neumann algebras. Subharmonic functions and their Riesz measures. Definition of the Brown measure. Brown measure of R-diagonal operators. 11.6.1 A little about the proof. 11.6.2 Example: circular operator. 11.6.3 The circular law. 11.6.4 The single ring theorem. 11.7 Brown measure of elliptic operators. 11.8 Brown measure for unbounded operators. 11.9 Hermitization method: using operator-valued free probability for calculating the Brown measure. 276 11.10 Brown measure of arbitrary polynomials in free variables . 266 267 268 271 272 273 273 274 275 275 Solutions to Exercises. to exercises in Chapter 1. to exercises in Chapter 2. to exercises in Chapter 3. to exercises in Chapter 4. to exercises in Chapter 5. to exercises in Chapter
6. to exercises in Chapter 7. to exercises in Chapter 8. to exercises in Chapter 9. to exercises in Chapter 10. to exercises in Chapter 11. 281 281 291 294 301 302 303 305 307 313 314 315 References. 319 Index of Exercises. 329 Index 331 11.3 11.4 11.5 11.6 12 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11 Solutions Solutions Solutions Solutions Solutions Solutions Solutions Solutions Solutions Solutions Solutions 277 |
| any_adam_object | 1 |
| author | Mingo, James A. Speicher, Roland 1960- |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics |
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| dewey-search | 510 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| spelling | Mingo, James A. Verfasser (DE-588)1171183291 aut Free probability and random matrices James A. Mingo, Roland Speicher New York, NY Springer [2017] © 2017 xiv, 336 Seiten Diagramme 23.5 cm x 15.5 cm txt rdacontent n rdamedia nc rdacarrier Fields Institute monographs Volume 35 Stochastische Matrix (DE-588)4057624-3 gnd rswk-swf Operatoralgebra (DE-588)4129366-6 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf PBT Cauchy transform Guassian random matrices asymptotic freeness free central limit theorem free harmonic analysis Wahrscheinlichkeitstheorie (DE-588)4079013-7 s Operatoralgebra (DE-588)4129366-6 s Stochastische Matrix (DE-588)4057624-3 s DE-604 Speicher, Roland 1960- Verfasser (DE-588)111876451 aut Springer Science + Business Media LLC (DE-588)1065492340 pbl Erscheint auch als Online-Ausgabe 978-1-4939-6942-5 (DE-604)BV044396774 Fields Institute monographs Volume 35 (DE-604)BV009737926 35 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3c8ec1e992864b4b9bf21f926df161b8&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029634584&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mingo, James A. Speicher, Roland 1960- Free probability and random matrices Fields Institute monographs Stochastische Matrix (DE-588)4057624-3 gnd Operatoralgebra (DE-588)4129366-6 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| subject_GND | (DE-588)4057624-3 (DE-588)4129366-6 (DE-588)4079013-7 |
| title | Free probability and random matrices |
| title_auth | Free probability and random matrices |
| title_exact_search | Free probability and random matrices |
| title_full | Free probability and random matrices James A. Mingo, Roland Speicher |
| title_fullStr | Free probability and random matrices James A. Mingo, Roland Speicher |
| title_full_unstemmed | Free probability and random matrices James A. Mingo, Roland Speicher |
| title_short | Free probability and random matrices |
| title_sort | free probability and random matrices |
| topic | Stochastische Matrix (DE-588)4057624-3 gnd Operatoralgebra (DE-588)4129366-6 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| topic_facet | Stochastische Matrix Operatoralgebra Wahrscheinlichkeitstheorie |
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