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This is a revised, updated and enlarged edition of Volume II of the author´s book "Spin Glasses: A Challenge for Mathematicians" designed to introduce this exciting work to the math-minded reader, in a rigorous manner requiring no knowledge of any physics.

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This is a new, completely revised, updated and enlarged edition of the author´s Ergebnisse vol. 46: "Spin Glasses: A Challenge for Mathematicians" in two volumes (this is the 2nd volume). In the eighties, a group of theoretical physicists introduced several models for certain disordered systems, called "spin glasses". These models are simple and rather canonical random structures, of considerable interest for several branches of science (statistical physics, neural networks and computer science). The physicists studied them by non-rigorous methods and predicted spectacular behaviors. This book introduces in a rigorous manner this exciting new area to the mathematically minded reader. It requires no knowledge whatsoever of any physics. The present Volume II contains a considerable amount of new material, in particular all the fundamental low-temperature results obtained after the publication of the first edition.

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Part I. Advanced Replica-Symmetry. - The Gardener Formula for the sphere. - The Gardener Formula for the Discrete Cube. - The Hopfield Model. - The SK Model Without External Field. - Part II. Low Temperature. - The Ghirlanda-Guerra Identities. - The High-Temperature Region of the SK Model. - The Parisi Formula. - The Parisi Solution. - The p -spin Interaction Model. - Appendix: Elements of Probability Theory. - References. - Index.

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From the reviews of the second edition:

"This is the second volume of the author´s treatise Mean field models for spin glasses ... . With its depth and profusion of important techniques, this book ... opens new perspectives in probability theory and statistical mechanics of disordered systems. It will be a source of inspiration for generations of mathematicians." (Francis Comets, Mathematical Reviews, November, 2013)
"This is the second volume of the treatise ´Mean fields models for spin glasses´ and it is truly the continuation of the preceding one. ... the book is a book of mathematics expanded around Hamiltonian. The main considered topics are the Gardner formula, the Hopfield model, the SK model, the Parisi formula and the p-spin interaction model. ... This is truly a nice book of condensed mathematics." (Guy Jumarie, Zentralblatt MATH, Vol. 1232, 2012)

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