Gibbs Measures and Phase Transitions

Description

A work of outstanding scholarship Covers in depth a broad range of topics in the mathematical theory of phase transition in statistical mechanics Informs about the state of the art in several directions Accessible to a general readership of mathematicians with a basic knowledge of measure theory and probability Does not assume any prior knowledge of statistical mechanics 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