Stochastic Processes on a Lattice and Gibbs Measures

1 — Aspects of the Ising Model.- 1. Introduction.- 2. Peieris’s Argument.- 3. Imposed Field, Thermodynamic Quantities.- 4. The Unidimensional Case and Tree.- 5. Antiferromagnetism.- 2 — Gibbs Measures.- 1. The DLR Problem.- 2. The Structure of G.- 3. Gibbs Specifications, Gibbs Measures.- 4. Relations with Thermodynamics.- 3 — The Existence of Gibbs Measures.- 1. Markovian Properties.- 2. Existence of Gibbs Measures when ? is Finite.- 3. Existence of Gibbs Measures — the General Case.- 4. Models of the P(?)-Type, Gaussian Gibbs Measures S.- 4 — Phase Transitions — 1: Methods of Convex Analysis.- 1. Holley’s Inequality.- 2. The FKG Inequality.- 3. Attracting Specifications.- 4. The Ising Model on ?2.- 5. Symmetry Breaking on ?d for d ? 3.- 5 — Other Inequalities.- 1. Urse11 Functions.- 2. GK S Inequalities.- 3. The GHS Inequality.- 4. The Simon-Lieb Inequality.- S. Additional Inequalities.- 6 — Phase Transitions — 2: Phase Diagrams and Perturbed Hamiltonians.- 1. Fundamental Configurations.- 2. Perturbed Hamiltonians.- 3. The Pirogov-Sinaï Theorem.- 4. Boundary Models.- S. Principle of the Proof.- 7 — Phase Transitions — 3: Positive Reflexivity.- 1. Reflexive Positivity.- 2. The Unidimensional Case.- 3. Checkerboard Estimate.- 4. Gaussian Domination.- 5. The Infrared Domination.- 8 — Continuous Symmetry and Other Methods.- 1. The Case of Continuous Symmetry.- 2. The Lack of Symmetry Breaking when d = 2.- 3. Spontaneous Magnetization when d ? 3.- 4. Onsager s Method.- 5. Combinatorial Approach.- 6. The Lee-Yang Theorem.- 9 — The Dynamics of Ising Systems.- 1. Introduction.- 2. A Finite Number of Sites.- 3. An Infinite Number of Sites.- 10 — Statistics and Applications.- 1. Ergodicity.- 2. Statistics.- 3. Image Processing.- 4. Other Applications.