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  3. Random Matrices: High Dimensional Phenomena
  • Cited by 27
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    • Publisher:
      Cambridge University Press
      Publication date:
      March 2012
      October 2009
      ISBN:
      9781139107129
      9780521133128
      DOI:
      https://doi.org/10.1017/CBO9781139107129
      Dimensions:
      Weight & Pages:
      Dimensions:
      (228 x 152 mm)
      Weight & Pages:
      0.63kg, 448 Pages
      • Subjects:
      Mathematics (general), Mathematics, Abstract Analysis, Probability Theory and Stochastic Processes, Statistics and Probability
      Series:
      London Mathematical Society Lecture Note Series (367)
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    Subjects:
    Mathematics (general), Mathematics, Abstract Analysis, Probability Theory and Stochastic Processes, Statistics and Probability
    Series:
    London Mathematical Society Lecture Note Series (367)
    Information

    Book description

    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.

    Reviews

    "The book under review is somewhat special in that it is not so much an introduction to the standard models and topics of random matrix theory, but rather to a set of functional analytic issues that are relevant to random matrices."
    Michael Stolz, Mathematical Reviews

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