J. A. Mingo and R. Speicher Free Probability and Random Matrices Schwinger-Verlag Saarbrucken¨ Kingston 2017 Dedicated to Dan-Virgil Voiculescu who gave us freeness and Betina and Jill who gave us the liberty to work on it. Contents 1 Asymptotic Freeness of Gaussian Random Matrices . 13 1.1 Moments and cumulants of random variables . 13 1.2 Moments of a Gaussian random variable . 15 1.3 Gaussian vectors . 16 1.4 The moments of a standard complex Gaussian random variable . 17 1.5 Wick’s formula . 17 1.6 Gaussian random matrices . 19 1.7 A genus expansion for the GUE . 19 1.8 Non-crossing partitions and permutations . 21 1.9 Wigner’s semi-circle law . 23 1.10 Asymptotic freeness of independent GUE’s . 23 1.11 Freeness and asymptotic freeness . 26 1.12 Basic properties of freeness . 26 1.13 Classical moment-cumulant formulas . 29 1.14 Additional exercises . 30 2 The Free Central Limit Theorem and Free Cumulants . 33 2.1 The classical and free central limit theorems . 33 2.1.1 Classical central limit theorem . 37 2.1.2 Free central limit theorem . 38 2.2 Non-crossing partitions and free cumulants . 41 2.3 Products of free random variables . 48 2.4 Functional relation between moment series and cumulant series . 49 2.5 Subordination and the non-commutative derivative . 52 3 Free Harmonic Analysis ........................................ 59 3.1 The Cauchy transform . 60 3.2 Moments and asymptotic expansions . 72 3.3 Analyticity of the R-transform: compactly supported measures . 74 3.4 Measures with finite variance . 78 5 6 Contents 3.5 Free additive convolution . 83 3.6 The R-transform and free additive convolution of arbitrary measures 88 4 Asymptotic Freeness ............................................ 99 4.1 Averaged convergence versus almost sure convergence . 99 4.2 Gaussian random matrices and deterministic matrices . 104 4.3 Haar distributed unitary random and deterministic matrices . 108 4.4 Wigner and deterministic random matrices . 112 4.5 Examples of random matrix calculations . 122 4.5.1 Wishart matrices and the Marchenko-Pastur distribution . 122 4.5.2 Sum of random matrices . 123 4.5.3 Product of random matrices . 124 5 Second-Order Freeness .........................................127 5.1 Fluctuations of GUE random matrices . 128 5.2 Fluctuations of several matrices . 138 5.3 second-order probability space and second-order freeness . 142 5.4 second-order cumulants . 147 5.5 Functional relation between moment and cumulant series . 154 5.6 Diagonalization of fluctuations . 156 5.6.1 Diagonalization in the one-matrix case . 157 5.6.2 Diagonalization in the multivariate case . 162 6 Free Group Factors and Freeness ................................165 6.1 Group (von Neumann) algebras . 165 6.2 Free group factors . 167 6.3 Free product of groups . 168 6.4 Moments and isomorphism of von Neumann algebras . 168 6.5 Freeness in the free group factors . 169 6.6 The structure of free group factors . 170 6.7 Compression of free group factors . 172 6.8 Circular operators and complex Gaussian random matrices . 173 6.9 Proof of (F3)1=2 (F9) .................................. 176 L 'L 6.10 The general case (Fn)1=k (F n k2 )..................... 178 L 'L 1+( 1) 6.11 Interpolating free group factors . .− . 178 6.12 The dichotomy for the free group factor isomorphism problem . 179 7 Free Entropy c- the Microstates Approach via Large Deviations . 181 7.1 Motivation . 181 7.2 Large deviation theory and Cramer’s´ Theorem . 182 7.3 Sanov’s Theorem and entropy . 186 7.4 Back to random matrices and one-dimensional free entropy . 188 7.5 Definition of multivariate free entropy . 190 7.6 Some important properties of c ............................... 192 Contents 7 7.7 Applications of free entropy to operator algebras . 194 7.7.1 The proof of Theorem 7, part (i)........................ 196 7.7.2 The proof of Theorem 7, part (iii)....................... 198 8 Free Entropy c∗- the Non-Microstates Approach via Free Fisher Information ...................................................201 8.1 Non-commutative derivatives . 201 8.2 ¶i as unbounded operator on C x1;:::;xn ...................... 204 h i 8.3 Conjugate variables and free Fisher information F∗ .............. 208 8.4 Additivity of F∗ and freeness . 216 8.5 The non-microstates free entropy c∗ ........................... 220 8.6 Operator algebraic applications of free Fisher information . 221 8.7 Absence of atoms for self-adjoint polynomials . 223 8.8 Additional exercises . 225 9 Operator-Valued Free Probability Theory and Block Random Matrices ......................................................229 9.1 Gaussian block random matrices . 229 9.2 General theory of operator-valued free probability . 238 9.3 Relation between scalar-valued and matrix-valued cumulants . 243 9.4 Moving between different levels . 245 9.5 A non-self-adjoint example . 248 10 Polynomials in Free Variables and Operator-Valued Convolution . 251 10.1 The general concept of a free deterministic equivalent . 251 10.2 A motivating example: reduction to multiplicative convolution . 253 10.3 Reduction to operator-valued additive convolution via the linearization trick. 255 10.4 Analytic theory of operator-valued convolutions . 259 10.4.1 General notations . 259 10.4.2 Operator-valued additive convolution . 260 10.4.3 Operator-valued multiplicative convolution. 261 10.5 Numerical Example. 261 10.6 The Case of Rational Functions . 262 10.7 Additional exercise . 264 11 Brown Measure ................................................265 11.1 Brown measure for normal operators . 265 11.2 Brown measure for matrices. 266 11.3 Fuglede-Kadison determinant in finite von Neumann algebras . 268 11.4 Subharmonic functions and their Riesz measures . 269 11.5 Definition of the Brown measure . 270 11.6 Brown measure of R-diagonal operators . 272 11.6.1 A little about the proof . 273 8 Contents 11.6.2 Example: circular operator . 274 11.6.3 The circular law . 274 11.6.4 The single ring theorem . 275 11.7 Brown measure of elliptic operators . 276 11.8 Brown measure for unbounded operators . 276 11.9 Hermitization method . 277 11.10Brown measure of arbitrary polynomials in free variables . 278 Solutions to Exercises ...............................................283 12.1 Solutions to exercises in Chapter 1 . 283 12.2 Solutions to exercises in Chapter 2 . 292 12.3 Solutions to exercises in Chapter 3 . 295 12.4 Solutions to exercises in Chapter 4 . 302 12.5 Solutions to exercises in Chapter 5 . 303 12.6 Solutions to exercises in Chapter 6 . 304 12.7 Solutions to exercises in Chapter 7 . 305 12.8 Solutions to exercises in Chapter 8 . 308 12.9 Solutions to exercises in Chapter 9 . 313 12.10Solutions to exercises in Chapter 10 . 314 12.11Solutions to exercises in Chapter 11 . 315 References .........................................................319 Index of Exercises ..................................................331 Introduction This book is an invitation to the world of free probability theory. Free probability is a quite young mathematical theory with many avatars. It owes its existence to the visions of one man, Dan-Virgil Voiculescu, who created it out of nothing at the beginning of the 1980s and pushed it forward ever since. The.