Preface -- Contents -- Introduction -- Part I. General theory and basic examples -- Chapter 1 Specifications of random fields -- 1.1 Preliminaries -- 1.2 Prescribing conditional probabilities -- 1.3 λ-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 -- 15.1 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.