Gibbs Random Fields Cluster Expansions
| Main Authors: | , |
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| Format: | eBook |
| Language: | English |
| Published: |
Dordrecht
Springer Netherlands
1991, 1991
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| Edition: | 1st ed. 1991 |
| Series: | Mathematics and its Applications, Soviet Series
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 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