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  • Book
  • © 2013

The Sherrington-Kirkpatrick Model

  • Presents many central ideas of the mathematical theory of the Sherrington-Kirkpartrick model in detail

  • Contains a fundamental breakthrough in this subject by the author

  • Accessible to graduate students working in probability theory or statistical mechanics

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

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eBook USD 109.00
Price excludes VAT (USA)
  • ISBN: 978-1-4614-6289-7
  • Instant PDF download
  • Readable on all devices
  • Own it forever
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  • Tax calculation will be finalised during checkout
Softcover Book USD 149.99
Price excludes VAT (USA)
Hardcover Book USD 149.99
Price excludes VAT (USA)

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Table of contents (4 chapters)

  1. Front Matter

    Pages i-xii
  2. The Free Energy and Gibbs Measure

    • Dmitry Panchenko
    Pages 1-31
  3. The Ruelle Probability Cascades

    • Dmitry Panchenko
    Pages 33-77
  4. The Parisi Formula

    • Dmitry Panchenko
    Pages 79-115
  5. Toward a Generalized Parisi Ansatz

    • Dmitry Panchenko
    Pages 117-135
  6. Back Matter

    Pages 137-156

About this book

The celebrated Parisi solution of the Sherrington-Kirkpatrick model for spin glasses is one of the most important achievements in the field of disordered systems. Over the last three decades, through the efforts of theoretical physicists and mathematicians, the essential aspects of the Parisi solution were clarified and proved mathematically. The core ideas of the theory that emerged are the subject of this book, including the recent solution of the Parisi ultrametricity conjecture and a conceptually simple proof of the Parisi formula for the free energy. The treatment is self-contained and should be accessible to graduate students with a background in probability theory, with no prior knowledge of spin glasses. The methods involved in the analysis of the Sherrington-Kirkpatrick model also serve as a good illustration of such classical topics in probability as the Gaussian interpolation and concentration of measure, Poisson processes, and representation results for exchangeable arrays.

Keywords

  • Aizenman-Sims-Starr scheme
  • Aldous-Hoover representation
  • Dovbysh-Sudakov representation
  • Gaussian processes
  • Ghirlanda-Guerra identities
  • Guerra replica symmetry breaking
  • Parisi ansatz
  • Parisi formula
  • Poisson processes
  • Poisson-Dirichlet processes
  • Ruelle probability cascades
  • Sherrington-Kirkpatrick model
  • Talagrand positivity principle
  • exchangeability
  • p-spin models
  • replica symmetry breaking
  • spin glass models
  • ultrametricity

Reviews

From the book reviews:

“The book is a very valuable addition to the literature in the field, and can be seen as complementary to other existing works … . I would recommend this book especially to young researchers. They can learn from it a lot about the scientific results, which could be extended to other important cases of hard optimization problems. It also provides an insight on developing original strategies when working with the difficult mathematical problems arising in studies of complex systems.” (Francesco Guerra, Bulletin of the American Mathematical Society, Vol. 52 (1), January, 2015)

“The monograph under review is concerned with the Sherrington-Kirkpatrick (SK) model of spin glasses. … The monograph deals with a notoriously difficult subject. Nevertheless, the author takes care to explain the ideas behind the proofs, which makes the text pleasant to read. It will most certainly remain a reference for years.” (José Trashorras, Mathematical Reviews, April, 2014)

Authors and Affiliations

  • , Department of Mathematics, Texas A&M University, College Station, USA

    Dmitry Panchenko

About the author

Dmitry Panchenko is Professor of Mathematics at Texas A&M University.

Bibliographic Information

Buying options

eBook USD 109.00
Price excludes VAT (USA)
  • ISBN: 978-1-4614-6289-7
  • 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 149.99
Price excludes VAT (USA)
Hardcover Book USD 149.99
Price excludes VAT (USA)