Our mathematical understanding of the statistical mechanics of disordered systems is going through a period of stunning progress. This self-contained book is a graduate-level introduction for mathematicians and for physicists interested in the mathematical foundations of the field, and can be used as a textbook for a two-semester course on mathematical statistical mechanics. It assumes only basic knowledge of classical physics and, on the mathematics side, a good working knowledge of graduate-level probability theory. The book starts with a concise introduction to statistical mechanics, proceeds to disordered lattice spin systems, and concludes with a presentation of the latest developments in the mathematical understanding of mean-field spin glass models. In particular, recent progress towards a rigorous understanding of the replica symmetry-breaking solutions of the Sherrington-Kirkpatrick spin glass models, due to Guerra, Aizenman-Sims-Starr and Talagrand, is reviewed in some detail.
Author(s): Anton Bovier
Series: Cambridge Series in Statistical and Probabilistic Mathematics
Publisher: Cambridge University Press
Year: 2006
Language: English
Pages: 324
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 11
Nomenclature......Page 15
Part I Statistical mechanics......Page 17
1 Introduction......Page 19
1.1 Thermodynamics......Page 20
2.1 The ideal gas in one dimension......Page 25
2.2 The micro-canonical ensemble......Page 29
2.3 The canonical ensemble and the Gibbs measure......Page 35
2.4 Non-ideal gases in the canonical ensemble......Page 38
2.5 Existence of the thermodynamic limit......Page 40
2.6 The liquid–vapour transition and the van der Waals gas......Page 44
2.7 The grand canonical ensemble......Page 47
3.1 Lattice gases......Page 49
3.2 Spin systems......Page 50
3.3 Subadditivity and the existence of the free energy......Page 52
3.4 The one-dimensional Ising model......Page 53
3.5 The Curie–Weiss model......Page 55
4.1 Spin systems and Gibbs measures......Page 65
4.2.1 Some topological background......Page 68
4.2.2 Local specifications and Gibbs measures......Page 70
4.3.1 Dobrushin’s uniqueness criterion......Page 75
4.3.2 The Peierls argument......Page 79
4.3.3 The FKG inequalities and monotonicity......Page 84
5.1 High-temperature expansions......Page 89
5.2 Polymer models: the Dobrushin–Koteck´y–Preiss criterion......Page 92
5.3 Convergence of the high-temperature expansion......Page 98
5.4.1 The Ising model at zero field......Page 104
5.4.2 Ground-states and contours......Page 105
Part II Disordered systems: lattice models......Page 111
6.1 Introduction......Page 113
6.2 Random Gibbs measures and metastates......Page 115
6.3 Remarks on uniqueness conditions......Page 122
6.4 Phase transitions......Page 123
6.5 The Edwards–Anderson model......Page 125
7.1 The Imry–Ma argument......Page 127
7.2.1 Translation-covariant states......Page 134
7.2.2 Order parameters and generating functions......Page 136
7.3 The Bricmont–Kupiainen renormalization group......Page 141
7.3.1 Renormalization group and contour models......Page 142
7.3.2 The ground-states......Page 147
7.3.3 The Gibbs states at finite temperature......Page 159
Part III Disordered systems: mean-field models......Page 175
8 Disordered mean-field models......Page 177
9.1 Ground-state energy and free energy......Page 181
9.2 Fluctuations and limit theorems......Page 185
9.3 The Gibbs measure......Page 191
9.4 The replica overlap......Page 196
9.5 Multi-overlaps and Ghirlanda–Guerra relations......Page 198
10.1 The standard GREM and Poisson cascades......Page 202
10.1.1 Poisson cascades and extremal processes......Page 203
10.1.2 Convergence of the partition function......Page 208
10.1.3 The Gibbs measures......Page 209
10.2 Models with continuous hierarchies: the CREM......Page 211
10.2.1 Free energy......Page 212
10.2.2 The empirical distance distribution......Page 218
10.2.3 Multi-overlap distributions......Page 219
10.3.1 Genealogy of flows of probability measures......Page 223
10.3.2 Coalescent processes......Page 225
10.3.3 Finite N setting for the CREM......Page 228
10.3.4 Neveu’s continuous state branching process......Page 230
10.3.5 Coalescence and Ghirlanda–Guerra identities......Page 232
11.1 The existence of the free energy......Page 234
11.2.1 Classical estimates on extremes......Page 236
11.3 The Parisi solution and Guerra’s bounds......Page 243
11.3.1 Application of a comparison lemma......Page 244
11.3.2 Computations with the GREM......Page 248
11.3.3 Talagrand’s theorem......Page 252
11.4 The Ghirlanda–Guerra relations in the SK models......Page 254
11.5 Applications in the p-spin SK model......Page 256
12.1 Origins of the model......Page 263
12.2 Basic ideas: finite M......Page 266
12.3 Growing M......Page 273
12.3.1 Fluctuations of Phi......Page 274
12.3.2 Logarithmic equivalence of lump-weights......Page 280
12.4 The replica symmetric solution......Page 281
12.4.1 Local convexity......Page 282
12.4.2 The cavity method 1......Page 287
12.4.3 Brascamp–Lieb inequalities......Page 289
12.4.4 The local mean field......Page 291
12.4.5 Gibbs measures and metastates......Page 294
12.4.6 The cavity method 2......Page 296
13.1 Number partitioning as a spin-glass problem......Page 301
13.2 An extreme value theorem......Page 303
13.3 Application to number partitioning......Page 305
References......Page 313
Index......Page 325
