Random Matrices: High Dimensional Phenomena (London Mathematical Society Lecture Note Series, Vol. 367) (London Mathematical Society Lecture Note Series, Series Number 367)

This Book Focuses On The Behaviour Of Large Random Matrices. Standard Results Are Covered, And The Presentation Emphasizes Elementary Operator Theory And Differential Equations, So As To Be Accessible To Graduate Students And Other Non-experts. The Introductory Chapters Review Material On Lie Groups And Probability Measures In A Style Suitable For Applications In Random Matrix Theory. Later Chapters Use Modern Convexity Theory To Establish Subtle Results About The Convergence Of Eigenvalue Distributions As The Size Of The Matrices Increases. Random Matrices Are Viewed As Geometrical Objects With Large Dimension. The Book Analyzes The Concentration Of Measure Phenomenon, Which Describes How Measures Behave On Geometrical Objects With Large Dimension. To Prove Such Results For Random Matrices, The Book Develops The Modern Theory Of Optimal Transportation And Proves The Associated Functional Inequalities Involving Entropy And Information. These Include The Logarithmic Sobolev Inequality, Which Measures How Fast Some Physical Systems Converge To Equilibrium. Metric Measure Spaces -- Lie Groups And Matrix Ensembles -- Entropy And Concentration Of Measure -- Free Entropy And Equilibrium -- Convergence To Equilibrium -- Gradient Flows And Functional Inequalities -- Young Tableaux -- Random Point Fields And Random Matrices -- Integrable Operators And Differential Equations -- Fluctuations And The Tracy-widom Distribution -- Limit Groups And Gaussian Measures -- Hermite Polynomials -- From The Ornstein-uhlenbeck Process To The Burgers Equation -- Noncommutative Probability Spaces. Gordon Blower. Includes Bibliographical References (p. 424-432) And Index. Title 5 Copyright 6 Dedication 7 Contents 9 Introduction 13 1 Metric measure spaces 16 Abstract 16 1.1 Weak convergence on compact metric spaces 16 1.2 Invariant measure on a compact metric group 22 1.3 Measures on non-compact Polish spaces 28 1.4 The Brunn-Minkowski inequality 34 1.5 Gaussian measures 37 1.6 Surface area measure on the spheres 39 1.7 Lipschitz functions and the Hausdorff metric 43 1.8 Characteristic functions and Cauchy transforms 45 2 Lie groups and matrix ensembles 54 Abstract 54 2.1 The classical groups, their eigenvalues and norms 54 2.2 Determinants and functional calculus 61 2.3 Linear Lie groups 68 2.4 Connections and curvature 75 2.5 Generalized ensembles 78 2.6 The Weyl integration formula 84 2.7 Dyson's circular ensembles 90 2.8 Circular orthogonal ensemble 93 2.9 Circular symplectic ensemble 95 3 Entropy and concentration of measure 96 Abstract 96 3.1 Relative entropy 96 3.2 Concentration of measure 105 3.3 Transportation 111 3.4 Transportation inequalities 115 3.5 Transportation inequalities for uniformlyconvex potentials 118 3.6 Concentration of measure in matrix ensembles 121 3.7 Concentration for rectangular Gaussian matrices 126 3.8 Concentration on the sphere 135 3.9 Concentration for compact Lie groups 138 4 Free entropy and equilibrium 144 Abstract 144 4.1 Logarithmic energy and equilibrium measure 144 4.2 Energy spaces on the disc 146 4.3 Free versus classical entropy on the spheres 154 4.4 Equilibrium measures for potentials on the real line 159 4.5 Equilibrium densities for convex potentials 166 4.6 The quartic model with positive leading term 171 4.7 Quartic models with negative leading term 176 4.8 Displacement convexity and relative free entropy 181 4.9 Toeplitz determinants 184 5 Convergence to equilibrium 189 Abstract 189 5.1 Convergence to arclength 189 5.2 Convergence of ensembles 191 5.3 Mean field convergence 195 5.4 Almost sure weak convergence for uniformly convex potentials 201 5.5 Convergence for the singular numbers from the Wishart distribution 205 6 Gradient flows and functional inequalities 208 Abstract 208 6.1 Variation of functionals and gradient flows 208 6.2 Logarithmic Sobolev inequalities 215 6.3 Logarithmic Sobolev inequalities for uniformlyconvex potentials 218 6.4 Fisher's information and Shannon's entropy 222 6.5 Free information and entropy 225 6.6 Free logarithmic Sobolev inequality 230 6.7 Logarithmic Sobolev and spectral gap inequalities 233 6.8 Inequalities for Gibbs measures onRiemannian manifolds 235 7 Young tableaux 239 Abstract 239 7.1 Group representations 239 7.2 Young diagrams 241 7.3 The Vershik distribution 249 7.4 Distribution of the longest increasing subsequence 255 7.5 Inclusion-exclusion principle 262 8 Random point fields and random matrices 265 Abstract 265 8.1 Determinantal random point fields 265 8.2 Determinantal random point fields on the real line 273 8.3 Determinantal random point fields and orthogonal polynomials 282 8.4 De Branges's spaces 286 8.5 Limits of kernels 290 9 Integrable operators and differential equations 293 Abstract 293 9.1 Integrable operators and Hankel integral operators 293 9.2 Hankel integral operators that commute with second order differential operators 301 9.3 Spectral bulk and the sine kernel 305 9.4 Soft edges and the Airy kernel 311 9.5 Hard edges and the Bessel kernel 316 9.6 The spectra of Hankel operators andrational approximation 322 9.7 The Tracy-Widom distribution 327 10 Fluctuations and the Tracy-Widom distribution 333 Abstract 333 10.1 The Costin-Lebowitz central limit theorem 333 10.2 Discrete Tracy-Widom systems 339 10.3 The discrete Bessel kernel 340 10.4 Plancherel measure on the partitions 346 10.5 Fluctuations of the longest increasing subsequence 355 10.6 Fluctuations of linear statistics over unitary ensembles 357 11 Limit groups and Gaussian measures 364 Abstract 364 11.1 Some inductive limit groups 364 11.1.2 Infinite torus 365 11.1.3 Infinite symmetric group 367 11.1.4 Infinite orthogonal group 367 11.2 Hua-Pickrell measure on the infinite unitary group 369 11.3 Gaussian Hilbert space 377 11.4 Gaussian measures and fluctuations 381 12 Hermite polynomials 385 Abstract 385 12.1 Tensor products of Hilbert space 385 12.2 Hermite polynomials and Mehler's formula 387 12.3 The Ornstein-Uhlenbeck semigroup 393 12.4 Hermite polynomials in higher dimensions 396 13 From the Ornstein-Uhlenbeck process to the Burgers equation 404 Abstract 404 13.1 The Ornstein-Uhlenbeck process 404 13.2 The logarithmic Sobolev inequality for the Ornstein-Uhlenbeck generator 408 13.3 The matrix Ornstein-Uhlenbeck process 410 13.4 Solutions for matrix stochastic differential equations 413 13.5 The Burgers equation 420 14 Noncommutative probability spaces 423 Abstract 423 14.1 Noncommutative probability spaces 423 14.2 Tracial probability spaces 426 14.3 The semicircular distribution 430 References 436 Index 445 Content: Cover; Title; Copyright; Dedication; Contents; Introduction; 1 Metric measure spaces; Abstract; 1.1 Weak convergence on compact metric spaces; 1.2 Invariant measure on a compact metric group; 1.3 Measures on non-compact Polish spaces; 1.4 The Brunn-Minkowski inequality; 1.5 Gaussian measures; 1.6 Surface area measure on the spheres; 1.7 Lipschitz functions and the Hausdorff metric; 1.8 Characteristic functions and Cauchy transforms; 2 Lie groups and matrix ensembles; Abstract; 2.1 The classical groups, their eigenvalues and norms; 2.2 Determinants and functional calculus. 2.3 Linear Lie groups2.4 Connections and curvature; 2.5 Generalized ensembles; 2.6 The Weyl integration formula; 2.7 Dyson's circular ensembles; 2.8 Circular orthogonal ensemble; 2.9 Circular symplectic ensemble; 3 Entropy and concentration of measure; Abstract; 3.1 Relative entropy; 3.2 Concentration of measure; 3.3 Transportation; 3.4 Transportation inequalities; 3.5 Transportation inequalities for uniformlyconvex potentials; 3.6 Concentration of measure in matrix ensembles; 3.7 Concentration for rectangular Gaussian matrices; 3.8 Concentration on the sphere. 3.9 Concentration for compact Lie groups4 Free entropy and equilibrium; Abstract; 4.1 Logarithmic energy and equilibrium measure; 4.2 Energy spaces on the disc; 4.3 Free versus classical entropy on the spheres; 4.4 Equilibrium measures for potentials on the real line; 4.5 Equilibrium densities for convex potentials; 4.6 The quartic model with positive leading term; 4.7 Quartic models with negative leading term; 4.8 Displacement convexity and relative free entropy; 4.9 Toeplitz determinants; 5 Convergence to equilibrium; Abstract; 5.1 Convergence to arclength; 5.2 Convergence of ensembles. 5.3 Mean field convergence5.4 Almost sure weak convergence for uniformly convex potentials; 5.5 Convergence for the singular numbers from the Wishart distribution; 6 Gradient flows and functional inequalities; Abstract; 6.1 Variation of functionals and gradient flows; 6.2 Logarithmic Sobolev inequalities; 6.3 Logarithmic Sobolev inequalities for uniformlyconvex potentials; 6.4 Fisher's information and Shannon's entropy; 6.5 Free information and entropy; 6.6 Free logarithmic Sobolev inequality; 6.7 Logarithmic Sobolev and spectral gap inequalities. 6.8 Inequalities for Gibbs measures onRiemannian manifolds7 Young tableaux; Abstract; 7.1 Group representations; 7.2 Young diagrams; 7.3 The Vershik distribution; 7.4 Distribution of the longest increasing subsequence; 7.5 Inclusion-exclusion principle; 8 Random point fields and random matrices; Abstract; 8.1 Determinantal random point fields; 8.2 Determinantal random point fields on the real line; 8.3 Determinantal random point fields and orthogonal polynomials; 8.4 De Branges's spaces; 8.5 Limits of kernels; 9 Integrable operators and differential equations; Abstract. Abstract: An introduction to the behaviour of random matrices. Suitable for postgraduate students and non-experts