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Random matrices : high dimensional phenomena /
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...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge ; New York :
Cambridge University Press,
2009.
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| Colección: | London Mathematical Society lecture note series ;
367. |
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- 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.