Gibbs Measures and Phase Transitions (eBook)

Preface 8
Contents 12
Introduction 16
Part I General theory and basic examples 24
Chapter 1 Specifications of random fields 26
Chapter 2 Gibbsian specifications 40
Chapter 3 Finite state Markov chains as Gibbs measures 59
Chapter 4 The existence problem 72
Chapter 5 Specifications with symmetries 96
Chapter 6 Three examples of symmetry breaking 109
Chapter 7 Extreme Gibbs measures 129
Chapter 8 Uniqueness 155
Chapter 9 Absence of symmetry breaking. Non- existence 183
Part II Markov chains and Gauss fields as Gibbs measures 204
Chapter 10 Markov fields on the integers I 205
Chapter 11 Markov fields on the integers II 224
Chapter 12 Markov fields on trees 253
Chapter 13 Gaussian fields 271
Part III Shift- invariant Gibbs measures 304
Chapter 14 Ergodicity 305
Chapter 15 The specific free energy and its minimization 323
Chapter 16 Convex geometry and the phase diagram 353
Part IV Phase transitions in reflection positive models 380
Chapter 17 Reflection positivity 382
Chapter 18 Low energy oceans and discrete symmetry breaking 397
Chapter 19 Phase transitions without symmetry breaking 423
Chapter 20 Continuous symmetry breaking in N- vector models 448
Bibliographical Notes 468
References 510
List of Symbols 554
Index 556