Introduction
Part Ⅰ. General theory and basic examples
Chapter 1 Specifications of random fields
1.1 Preliminaries
1.2 Prescribing conditional probabilities
1.3 A-specifications
Chapter 2 Gibbsian specifications
2.1 Potentials
2.2 Quasilocality
2.3 Gibbs representation of pre-modifications
2.4 Equivalence of potentials
Chapter 3 Finite state Markov chains as Gibbs measures
3.1 Markov specifications on the integers
3.2 The one-dimensional Ising model
3.A Appendix. Positive matrices
Chapter 4 The existence problem
4.1 Local convergence of random fields
4.2 Existence of cluster points
4.3 Continuity results
4.4 Existence and topological properties of Gibbs measures
4.A Appendix. Standard Borel spaces
Chapter 5 Specifications with symmetries
5.1 Transformations of specifications
5.2 Gibbs measures with symmetries
Chapter 6 Three examples of symmetry breaking
6.1 Inhomogeneous Ising chains
6.2 The Ising ferromagnet in two dimensions
6.3 Shlosman's random staircases
Chapter 7 Extreme Gibbs measures
7.1 Tail triviality and approximation
7.2 Some applications
7.3 Extreme decomposition
7.4 Macroscopic equivalence of Gibbs simplices
Chapter 8 Uniqueness
8.1 Dobrushin's condition of weak dependence
8.2 Further consequences of Dobrushin's condition
8.3 Uniqueness in one dimension
Chapter 9 Absence of symmetry breaking. Non-existence
9.1 Discrete symmetries in one dimension
9.2 Continuous symmetries in two dimensions
Part Ⅱ. Markov chains and Gauss fields as Gibbs measures
Chapter 10 Markov fields on the integers I
10.1 Two-sided and one-sided Markov property
10.2 Markov fields which are Markov chains
10.3 Uniqueness of the shift-invariant Markov field
Chapter 11 Markov fields on the integers II
11.1 Boundary laws, uniqueness, and non-existence
11.2 The Spitzer-Cox example of phase transition
11.3 Kalikow's example of phase transition
11.4 Spitzer's example of totally broken shift-invariance
Chapter 12 Markov fields on trees
12.1 Markov chains and boundary laws
12.2 The Ising model on Cayley trees
Chapter 13 Gaussian fields
13.1 Gauss fields as Gibbs measures
13.2 Gibbs measures for Gaussian specifications
13.3 The homogeneous case
13.A Appendix. Some tools of Gaussian analysis
Part Ⅲ. Shift-invariant Gibbs measures
Chapter 14 Ergodicity
14.1 Ergodic random fields
14.2 Ergodic Gibbs measures
14.A Appendix. The multidimensional ergodic theorem
Chapter 15 The specific free energy and its minimization
I5.I Relative entropy
15.2 Specific entropy
15.3 Specific energy and free energy
15.4 The variational principle
15.5 Large deviations and equivalence of ensembles
Chapter 16 Convex geometry and the phase diagram
16.1 The pressure and its tangent functionals
16.2 A geometric view of Gibbs measures
16.3 Phase transitions with prescribed order parameters
16.4 Ubiquity of pure phases
Part Ⅳ. Phase transitions in reflection positive models
Chapter 17 Reflection positivity
17.1 The chessboard estimate
17.2 Gibbs distributions with periodic boundary condition
Chapter 18 Low energy oceans and discrete symmetry breaking
18.1 Percolation of spin patterns
18.2 Discrete symmetry breaking at low temperatures
18.3 Examples
Chapter 19 Phase transitions without symmetry breaking
19.1 Potentials with degenerated ground states, and perturbations thereof
19.2 Exploiting Sperner's lemma
19.3 Models with an entropy-energy conflict
19.A Appendix. Sperner's lemma
Chapter 20 Continuous symmetry breaking in N-vector models
20.1 Some preliminaries
20.2 Spin wave analysis, and spontaneous magnetization
Bibliographical Notes
Further Progress
References
References to the Second Edition
List of Symbols
Index
