Gibbs Random Fields: Cluster Expansions: Mathematics and its Applications, cartea 44

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-Itô-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 Möbius 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.