Gibbs Random Fields: Cluster Expansions

'Et moi, ..., si j' avait su comment en revenir, One service mathematics has rendered the human race. It has put common sense back je n'y serais point aIle.' Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense'" able 10 do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound_ Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com- puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

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.