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  • Book
  • © 2014

Superconcentration and Related Topics

  • First book devoted to the topic of super concentration, chaos and multiple valleys

  • Presents a wide array of examples on the subject

  • Integrates new concepts and gives a systematic account of the history and development of the features of random objects

  • Includes supplementary material: sn.pub/extras

Part of the book series: Springer Monographs in Mathematics (SMM)

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eBook USD 79.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-03886-5
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  • Own it forever
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  • Tax calculation will be finalised during checkout
Softcover Book USD 99.99
Price excludes VAT (USA)
Hardcover Book USD 139.99
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Table of contents (12 chapters)

  1. Front Matter

    Pages I-IX
  2. Introduction

    • Sourav Chatterjee
    Pages 1-14
  3. Markov Semigroups

    • Sourav Chatterjee
    Pages 15-22
  4. Superconcentration and Chaos

    • Sourav Chatterjee
    Pages 23-31
  5. Multiple Valleys

    • Sourav Chatterjee
    Pages 33-43
  6. Independent Flips

    • Sourav Chatterjee
    Pages 63-71
  7. Extremal Fields

    • Sourav Chatterjee
    Pages 73-85
  8. Further Applications of Hypercontractivity

    • Sourav Chatterjee
    Pages 87-103
  9. The Interpolation Method for Proving Chaos

    • Sourav Chatterjee
    Pages 105-114
  10. Variance Lower Bounds

    • Sourav Chatterjee
    Pages 115-123
  11. Dimensions of Level Sets

    • Sourav Chatterjee
    Pages 125-136
  12. Back Matter

    Pages 137-156

About this book

A certain curious feature of random objects, introduced by the author as “super concentration,” and two related topics, “chaos” and “multiple valleys,” are highlighted in this book. Although super concentration has established itself as a recognized feature in a number of areas of probability theory in the last twenty years (under a variety of names), the author was the first to discover and explore its connections with chaos and multiple valleys. He achieves a substantial degree of simplification and clarity in the presentation of these findings by using the spectral approach.

Understanding the fluctuations of random objects is one of the major goals of probability theory and a whole subfield of probability and analysis, called concentration of measure, is devoted to understanding these fluctuations. This subfield offers a range of tools for computing upper bounds on the orders of fluctuations of very complicated random variables. Usually, concentration of measure is useful when more direct problem-specific approaches fail; as a result, it has massively gained acceptance over the last forty years. And yet, there is a large class of problems in which classical concentration of measure produces suboptimal bounds on the order of fluctuations. Here lies the substantial contribution of this book, which developed from a set of six lectures the author first held at the Cornell Probability Summer School in July 2012.

The book is interspersed with a sizable number of open problems for professional mathematicians as well as exercises for graduate students working in the fields of probability theory and mathematical physics. The material is accessible to anyone who has attended a graduate course in probability.

Keywords

  • 60E15, 60K35, 60G15, 82B44, 60G60, 60G70
  • Gaussian field
  • chaos
  • concentration of measure
  • disordered system
  • hypercontractivity
  • multiple valleys
  • spin glass
  • superconcentration

Authors and Affiliations

  • Department of Statistics, Stanford University, Stanford, USA

    Sourav Chatterjee

About the author

Sourav Chatterjee is a Professor of Statistics and Mathematics at Stanford University. He has previously taught at the University of California at Berkeley and at the Courant Institute of Mathematical Sciences. He has won several international awards for his work in probability theory, including the Rollo Davidson Prize (2010), the Doeblin Prize (2012) and the Loève Prize (2013) and has received the invitation to speak at the International Congress of Mathematicians in 2014.

Bibliographic Information

Buying options

eBook USD 79.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-03886-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 99.99
Price excludes VAT (USA)
Hardcover Book USD 139.99
Price excludes VAT (USA)