| Description |
1 online resource (x, 437 pages). |
| Physical Medium |
polychrome |
| Description |
text file |
| Series |
London Mathematical Society lecture note series ; 367
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London Mathematical Society lecture note series ; 367.
|
| Summary |
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. |
| Bibliography |
Includes bibliographical references (pages 424-432) and index. |
| Contents |
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. |
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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. |
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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. |
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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. |
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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. |
| Local Note |
eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - North America |
| Subject |
Random matrices.
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Random matrices. |
| Genre/Form |
Electronic books.
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| Other Form: |
Print version: Blower, G. (Gordon). Random matrices. Cambridge ; New York : Cambridge University Press, ©2009 9780521133128 (DLC) 2010275269 (OCoLC)401146699 |
| ISBN |
9781139127547 (electronic book) |
|
1139127543 (electronic book) |
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9781139107129 (electronic book) |
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1139107127 (electronic book) |
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9780521133128 (paperback) |
|
0521133122 (paperback) |
|
9781139114714 |
|
1139114719 |
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9781283295864 |
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1283295865 |
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