Presenting and developing the theory of spin glasses as a prototype for complex systems, this book is a rigorous and up-to-date introduction to their properties. The book combines a mathematical description with a physical insight of spin glass models. Topics covered include the physical origins of those models and their treatment with replica theory; mathematical properties like correlation inequalities and their use in the thermodynamic limit theory; main exact solutions of the mean field models and their probabilistic structures; and the theory of the structural properties of the spin glass phase such as stochastic stability and the overlap identities. Finally, a detailed account is given of the recent numerical simulation results and properties, including overlap equivalence, ultrametricity and decay of correlations. The book is ideal for mathematical physicists and probabilists working in disordered systems.

Preface vii

1 Origins, models and motivations 1

1.1 The spi n glass problem 1

1.2 Random interactions, fi nite-dimensional models, mean-field models 2

1.3 Quenched measure and real replicas 6

1.4 Definition of a mean-field spin glass 9

1.5 Repl ica method for the SK model 13

1.6 The replica symmetric solution 16

1.7 The ultrametric replica symmetry breaking solution 19

2 Correlation inequalities 27

2.1 Spin functions 27

2.2 Ferromagnetism and the Griffiths-Kelly-Sherm an inequalities 31

2.3 Spin glass correlation inequalities of type I 34

2.4 Extension to quantum case 35

2.5 Type II inequalities: results and cou nterexamples 37

2.6 Inequalities of types I and II on the Nishimori line 39

2.7 Some consequences of correlation inequalities 44

3 The infinite-volume limt 47

3.1 Introduction 47

3.2 Finite-dimensional models with Gaussian interactions 48

3.3 Pressure self-averaging 53

3.4 Finite-dimensional models with centered interactions 56

3.5 Quantum models 59

3.6 Extension to non-centered interactions 62

3.7 Independence of the pressure from boundary conditions 64

3.8 Surface pressure 66

3.9 Complete theory on the Nishimori line 73

3.10 Thermodynamical limit for mean-field models, REM and OREM 78

4 Exact results for mean-field models 84

4.1 Introduction to the Poisson point process 84

4.2 Poisson poi nt process with exponential intensity 87

4.3 The Poisson-Dirichlet distribution 91

4.4 The random energy model 97

4.5 Upper bound for the REM pressure 98

4.6 Lower bound for t he REM pressure 103

4.7 Statistics of energy levels for the REM 107

4.8 The generalized random energy model 108

4.9 Upper bound for the GREM pressure 109

4.10 Lower bound for the OREM pressure 112

4.11 The Aizenman-Sims-Srarr extended variational pri11ciple 113

4.12 Guerra upper bound 119

4.13 Talagrand theorem and open problems 123

5 Spin glass identities 125

5.1 The stability method and the structural identities 125

5.2 Stochastic stabi lity identities 130

5.3 Strong stochastic stability and marginal independence 134

5.4 Graph theoretica l approach to A izenman-Contucci identities 137

5.5 Identities from self-averaging 144

5.6 Identities on the Nishimori line 146

5.7 Interaction flip identities 151

6 Numerical simulations 164

6.1 Introduction 164

6.2 Simulations with real replicas 165

6.3 Overlap equivalence 168

6.4 Ultrametricity l76

6.5 Decay of correlations 183

6.6 Energy interfaces 192

References 200

Index 209

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