In many domains one encounters "systems" of interacting elements, elements that interact more forcefully the closer they may be. The historical example upon which the theory offered in this book is based is that of magnetization as it is described by the Ising model. At the vertices of a regular lattice of sites, atoms "choos e" an orientation under the influence of the orientations of the neighboring atoms. But other examples are known, in physics (the theories of gasses, fluids, .. J, in biology (cells are increasingly likely to become malignant when their neighboring cells are malignant), or in medecine (the spread of contagious deseases, geogenetics, .. .), even in the social sciences (spread of behavioral traits within a population). Beyond the spacial aspect that is related to the idea of "neighboring" sites, the models for all these phenomena exhibit three common features: - The unavoidable ignorance about the totality of the phenomenon that is being studied and the presence of a great number of often unsuspected factors that are always unquantified lead inevitably to stochastic models. The concept of accident is very often inherent to the very nature of the phenomena considered, so, to justify this procedure, one has recourse to the physicist's principle of indeterminacy, or, for example, to the factor of chance in the Mendelian genetics of phenotypes.

Inhalt
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.