Mathematics and Its Applications

1. Gibbs Fields (Basic Notions).- x0 First Acquaintance with Gibbs Fields.- x1 Gibbs Modifications.- x2 Gibbs Modifications under Boundary Conditions and Definition of Gibbs Fields by Means of Conditional Distributions.- 2. Semi-Invariants and Combinatorics.- x1 Semi-Invariants and Their Elementary Properties.- x2 Hermite-Ito-Wick Polynomials. Diagrams. Integration by Parts.- x3 Estimates on Moments and Semi-Invariants of Functional of Gaussian Families.- x4 Connectedness and Summation over Trees.- x5 Estimates on Intersection Number.- x6 Lattices and Computations of Their Mobius Functions.- x7 Estimate of Semi-Invariants of Partially Dependent Random Variables.- x8 Abstract Diagrams (Algebraic Approach).- 3. General Scheme of Cluster Expansion.- x1 Cluster Representation of Partition Functions and Ensembles of Subsets.- x2 Cluster Expansion of Correlation Functions.- x3 Limit Correlation Function and Cluster Expansion of Measures.- x4 Cluster Expansion and Asymptotics of Free Energy. Analyticity of Correlation Functions.- x5 Regions of Cluster Expansions for the Ising Model.- x6 Point Ensembles.- 4. Small Parameters in Interactions.- x1 Gibbs Modifications of Independent Fields with Bounded Potential.- x2 Unbounded Interactions in the Finite-Range Part of a Potential.- x3 Gibbs Modifications of d-Dependent Fields.- x4 Gibbs Point Field in Rv.- x5 Models with Continuous Time.- x6 Expansion of Semi-Invariants. Perturbation of a Gaussian Field.- x7 Perturbation of a Gaussian Field with Slow Decay of Correlations.- x8 Modifications of d-Markov Gaussian Fields (Interpolation of Inverse Covariance).- 5. Expansions Around Ground States (Low-Temperature Expansions).- x1 Discrete Spin: Countable Number of Ground States.- x2 Continuous Spin: Unique Ground State.- x3 Continuous Spin: Two Ground States.- 6. Decay of Correlations.- x1 Hierarchy of the Properties of Decay of Correlations.- x2 An Analytic Method of Estimation of Semi-Invariants of Bounded Quasi-Local Functionals.- x3 A Combinatorial Method of Estimation of Semi-Invariants in the Case of Exponentially-Regular Cluster Expansion.- x4 Slow (power) Decay of Correlations.- x5 Low-Temperature Region.- x6 Scaling Limit of a Random Field.- 7. Supplementary Topics and Applications.- x1 Gibbs Quasistates.- x2 Uniqueness of Gibbs Fields.- x3 Compactness of Gibbs Modifications.- x4 Gauge Field with Gauge Group Z2.- x5 Markov Processes with Local Interaction.- x6 Ensemble of External Contours.- Concluding Remarks.- Bibliographic Comments.- References.