Random matrices: high dimensional phenomena
by Blower, G.
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 dist…
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- Full title :Random matrices : high dimensional phenomena / Gordon Blower.
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.
Read More ↓Edition Statements
- Publication Date:2009
Publication Statements
- Publisher(s):Cambridge University Press
- Place of Publication:Cambridge
- Language:eng
- Publication Date:2009
Notes
Title from publisher's bibliographic system (viewed on 05 Oct 2015).
The Physical Object
- Description:1 online resource (x, 437 Seiten)
Edition Identifiers
- ISBN:9781139107129, 1139107127, 9780521133128, 0521133122
- DOI:10.1017/CBO9781139107129
- oclc:ocn992903621
- LC Classification:QA188
- Dewey Decimal:512.9434
Table of Contents
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.
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