Gibbs Random Fields

Inhaltsverzeichnis

• 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.