عنوان

Gibbs Random Fields Cluster Expansions.

9401137080

9789401137089

b595275

Gibbs Random Fields Cluster Expansions.

[Book]

Malyshev, V.A.

Springer Verlag

2012

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-Ito-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 Moebius 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: Unique Ground 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.

Malyshev, V.A.

Malyshev, V.A.

Minlos, Robert A.

[Book]

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