Gibbs Random Fields - Robert A. Minlos

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Résumé :
1. Gibbs Fields (Basic Notions).- ?0 First Acquaintance with Gibbs Fields.- ?1 Gibbs Modifications.- ?2 Gibbs Modifications under Boundary Conditions and Definition of Gibbs Fields by Means of Conditional Distributions.- 2. Semi-Invariants and Combinatorics.- ?1 Semi-Invariants and Their Elementary Properties.- ?2 Hermite-It?-Wick Polynomials. Diagrams. Integration by Parts.- ?3 Estimates on Moments and Semi-Invariants of Functional of Gaussian Families.- ?4 Connectedness and Summation over Trees.- ?5 Estimates on Intersection Number.- ?6 Lattices and Computations of Their M?bius Functions.- ?7 Estimate of Semi-Invariants of Partially Dependent Random Variables.- ?8 Abstract Diagrams (Algebraic Approach).- 3. General Scheme of Cluster Expansion.- ?1 Cluster Representation of Partition Functions and Ensembles of Subsets.- ?2 Cluster Expansion of Correlation Functions.- ?3 Limit Correlation Function and Cluster Expansion of Measures.- ?4 Cluster Expansion and Asymptotics of Free Energy. Analyticity of Correlation Functions.- ?5 Regions of Cluster Expansions for the Ising Model.- ?6 Point Ensembles.- 4. Small Parameters in Interactions.- ?1 Gibbs Modifications of Independent Fields with Bounded Potential.- ?2 Unbounded Interactions in the Finite-Range Part of a Potential.- ?3 Gibbs Modifications of d-Dependent Fields.- ?4 Gibbs Point Field in Rv.- ?5 Models with Continuous Time.- ?6 Expansion of Semi-Invariants. Perturbation of a Gaussian Field.- ?7 Perturbation of a Gaussian Field with Slow Decay of Correlations.- ?8 Modifications of d-Markov Gaussian Fields (Interpolation of Inverse Covariance).- 5. Expansions Around Ground States (Low-Temperature Expansions).- ?1 Discrete Spin: Countable Number of Ground States.- ?2 Continuous Spin: UniqueGround State.- ?3 Continuous Spin: Two Ground States.- 6. Decay of Correlations.- ?1 Hierarchy of the Properties of Decay of Correlations.- ?2 An Analytic Method of Estimation of Semi-Invariants of Bounded Quasi-Local Functionals.- ?3 A Combinatorial Method of Estimation of Semi-Invariants in the Case of Exponentially-Regular Cluster Expansion.- ?4 Slow (power) Decay of Correlations.- ?5 Low-Temperature Region.- ?6 Scaling Limit of a Random Field.- 7. Supplementary Topics and Applications.- ?1 Gibbs Quasistates.- ?2 Uniqueness of Gibbs Fields.- ?3 Compactness of Gibbs Modifications.- ?4 Gauge Field with Gauge Group Z2.- ?5 Markov Processes with Local Interaction.- ?6 Ensemble of External Contours.- Concluding Remarks.- Bibliographic Comments.- References....

Sommaire:
1. Gibbs Fields (Basic Notions).- ?0 First Acquaintance with Gibbs Fields.- ?1 Gibbs Modifications.- ?2 Gibbs Modifications under Boundary Conditions and Definition of Gibbs Fields by Means of Conditional Distributions.- 2. Semi-Invariants and Combinatorics.- ?1 Semi-Invariants and Their Elementary Properties.- ?2 Hermite-It?-Wick Polynomials. Diagrams. Integration by Parts.- ?3 Estimates on Moments and Semi-Invariants of Functional of Gaussian Families.- ?4 Connectedness and Summation over Trees.- ?5 Estimates on Intersection Number.- ?6 Lattices and Computations of Their M?bius Functions.- ?7 Estimate of Semi-Invariants of Partially Dependent Random Variables.- ?8 Abstract Diagrams (Algebraic Approach).- 3. General Scheme of Cluster Expansion.- ?1 Cluster Representation of Partition Functions and Ensembles of Subsets.- ?2 Cluster Expansion of Correlation Functions.- ?3 Limit Correlation Function and Cluster Expansion of Measures.- ?4 Cluster Expansion and Asymptotics of Free Energy. Analyticity of Correlation Functions.- ?5 Regions of Cluster Expansions for the Ising Model.- ?6 Point Ensembles.- 4. Small Parameters in Interactions.- ?1 Gibbs Modifications of Independent Fields with Bounded Potential.- ?2 Unbounded Interactions in the Finite-Range Part of a Potential.- ?3 Gibbs Modifications of d-Dependent Fields.- ?4 Gibbs Point Field in Rv.- ?5 Models with Continuous Time.- ?6 Expansion of Semi-Invariants. Perturbation of a Gaussian Field.- ?7 Perturbation of a Gaussian Field with Slow Decay of Correlations.- ?8 Modifications of d-Markov Gaussian Fields (Interpolation of Inverse Covariance).- 5. Expansions Around Ground States (Low-Temperature Expansions).- ?1 Discrete Spin: Countable Number of Ground States.- ?2 Continuous Spin: UniqueGround State.- ?3 Continuous Spin: Two Ground States.- 6. Decay of Correlations.- ?1 Hierarchy of the Properties of Decay of Correlations.- ?2 An Analytic Method of Estimation of Semi-Invariants of Bounded Quasi-Local Functionals.- ?3 A Combinatorial Method of Estimation of Semi-Invariants in the Case of Exponentially-Regular Cluster Expansion.- ?4 Slow (power) Decay of Correlations.- ?5 Low-Temperature Region.- ?6 Scaling Limit of a Random Field.- 7. Supplementary Topics and Applications.- ?1 Gibbs Quasistates.- ?2 Uniqueness of Gibbs Fields.- ?3 Compactness of Gibbs Modifications.- ?4 Gauge Field with Gauge Group Z2.- ?5 Markov Processes with Local Interaction.- ?6 Ensemble of External Contours.- Concluding Remarks.- Bibliographic Comments.- References.