Lecture Notes in Mathematics

By: Erwin Bolthausen (Editor), Anton Bovier (Editor)

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Paperback | 11 December 2006

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Spin glass theory is going through a stunning period of progress while finding exciting new applications in areas beyond theoretical physics, in particular in combinatorics and computer science. This collection of state-of-the-art review papers written by leading experts in the field covers the topic from a wide variety of angles. The topics covered are mean field spin glasses, including a pedagogical account of Talagrand's proof of the Parisi solution, short range spin glasses, emphasizing the open problem of the relevance of the mean-field theory for lattice models, and the dynamics of spin glasses, in particular the problem of ageing in mean field models.

The book will serve as a concise introduction to the state of the art of spin glass theory, useful to both graduate students and young researchers, as well as to anyone curious to know what is going on in this exciting area of mathematical physics.

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You Can Find This Book In

Non-Fiction Science Physics Materials & States of Matter Condensed Matter Physics including Liquid State & Solid State Physics Mathematics Probability & Statistics Statistical Physics Algebra Reference, Information & Interdisciplinary Subjects Research & Information Information theory

Cybernetics & Systems Theory Mathematical Physics

Random Media and Spin Glasses: An Introduction into Some Mathematical Results and Problems	p. 1
Introduction	p. 1
The Basic Mathematical Models	p. 3
The Sherrington-Kirkpatrick Model	p. 6
Basic Properties of the SK-Model	p. 7
The Replica Computation	p. 9
The Parisi Ansatz	p. 12
Mathematically Rigorous Results for the SK-Model	p. 16
Guerra's Interpolation Method	p. 16
Mathematical Variants of the Cavity Method	p. 20
The Random Energy Models	p. 23
The REM	p. 23
The Generalized Random Energy Model	p. 26
The Aizenman-Sims-Starr Proof of Guerra's Bound of the SK Free Energy	p. 30
Some Open Problems: Ultrametricity and Chaos	p. 35
Other Models: Some Comments about the Perceptron	p. 39
Dynamic Problems and Aging	p. 41
References	p. 43
Spin Glasses: A Perspective	p. 45
Introduction	p. 45
Random Magnetic Alloys	p. 45
More Details	p. 47
Discontinuous Transitions	p. 55
Beyond Magnetic Alloys	p. 56
Complex Many-Body Problems	p. 56
Dynamics	p. 58
Examples	p. 59
Conclusion	p. 60
Acknowledgements	p. 61
References	p. 61
Mean Field Models for Spin Glasses: Some Obnoxious Problems	p. 63
Introduction	p. 63
Guerra's Replica-Symmetric Bound	p. 66
Guerra's Broken Replica-Symmetry Bound	p. 70
The Ultrametricity Conjecture	p. 75
The Chaos Problem	p. 77
References	p. 80
Much Ado about Derrida's GREM	p. 81
Introduction	p. 81
History of the Models	p. 82
Geometry of Gibbs Measures	p. 85
Point Process of Extremes	p. 87
Organisation of the Paper	p. 88
Convergent Point Processes Associated to the GREM	p. 88
GREM: Detailed Analysis	p. 92
Fluctuations of the Partition Function	p. 92
Gibbs Measure: Approach via Ruelle's Probability Cascades	p. 04
Distribution of the Overlaps	p. 95
Ghirlanda-Guerra Identities	p. 96
Empirical Distance Distribution Function [Kappa subscript beta]	p. 99
CREM: Implicit Approach	p. 99
Maximum of the Hamiltonian. Limit of the Free Energy	p. 99
Two-Replicas Ultrametric Distance Distribution Function	p. 101
Ghirlanda-Guerra Identities	p. 101
Marginals of [Kappa subscript beta] in Terms of Ruelle's Probability Cascades	p. 102
Genealogies and Neveu's Branching Process	p. 103
Problems with the Explicit Description of Limiting Gibbs Measures	p. 103
Genealogical Map of a Flow of Probability Measures	p. 104
Coalescent Associated with a Flow of Probability Measures	p. 106
Finite N Setting for the CREM	p. 106
Genealogy of a Continuous State Branching Process	p. 108
Main Result	p. 109
Coalescence and Ghirlanda-Guerra identities	p. 110
References	p. 113
Dynamics for Spherical Models of Spin-Glass and Aging	p. 117
Introduction	p. 117
Spherical Sherrington-Kirkpatrick Model	p. 121
Spherical p-spins Model	p. 132
Future problems	p. 141
References	p. 142
Local vs. Global Variables for Spin Glasses	p. 145
Local Variables and Thermodynamic Limits	p. 145
Nearest-Neighbor Ising Ferromagnets	p. 148
Finite-Range and Infinite-Range Spin Glasses	p. 149
A New Global Order Parameter for Spin Glasses?	p. 151
What are Metastates, Anyway?	p. 155
References	p. 157
Short-Range Spin Glasses: Results and Speculations	p. 159
Introduction	p. 160
The Trinity of Scenarios	p. 161
Heuristic Description	p. 162
Description via the Metastate	p. 164
Replica Symmetry Breaking for Short-Range Models	p. 166
Wild Possibilities	p. 168
Phase Transition Without Broken Spin-Flip Symmetry	p. 169
Highly Chaotic Temperature Dependence	p. 172
References	p. 174
Index	p. 177
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Lecture Notes in Mathematics

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