The Sherrington-Kirkpatrick Model

ISBN-13: 9781489993731 / Angielski / Miękka / 2015 / 156 str.

Dmitry Panchenko

Opis

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.

Recenzje

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)

Spis treści

Preface.- 1 The Free Energy and Gibbs Measure.- 2 The Ruelle Probability Cascades.- 3 The Parisi Formula.- 4 Toward a Generalized Parisi Ansatz.- A Appendix.- Bibliography.- Notes and Comments.- References.- Index.

O autorze

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