Abstract cluster expansion with applications to statistical mechanical systems

@article{Poghosyan2008AbstractCE,
  title={Abstract cluster expansion with applications to statistical mechanical systems},
  author={Suren Poghosyan and Daniel Ueltschi},
  journal={Journal of Mathematical Physics},
  year={2008},
  volume={50},
  pages={053509}
}
We formulate a general setting for the cluster expansion method and we discuss sufficient criteria for its convergence. We apply the results to systems of classical and quantum particles with stable interactions. 
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51 Citations
Highly Influential Citations
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Background Citations
22
Methods Citations
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51 Citations

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References

SHOWING 1-10 OF 47 REFERENCES
Cluster expansions and correlation functions
A cluster expansion is proposed, that applies to both continuous and discrete systems. The assumption for its convergence involves an extension of the neat Kotecky-Preiss criterion. Expressions and…
On the convergence of cluster expansions
A new form of the Mayer expansion in classical statistical mechanics
New expressions are given for the expansion coefficients in the Mayer expansion (and thus the virial expansion). These promise to be useful in applications, as well as provide a simple rigorous proof…
Cluster expansion for abstract polymer models
A new direct proof of convergence of cluster expansions for polymer (contour) models is given in an abstract setting. It does not rely on Kirkwood-Salsburg type equations or “combinatorics of trees.”…
Abstract Polymer Models with General Pair Interactions
Abstract A convergence criterion of cluster expansion is presented in the case of an abstract polymer system with general pair interactions (i.e. not necessarily hard core or repulsive). As a…
A Simple Inductive Approach to the Problem of Convergence of Cluster Expansions of Polymer Models
We explain a simple inductive method for the analysis of the convergence of cluster expansions (Taylor expansions, Mayer expansions) for the partition functions of polymer models. We give a very…
Cluster Expansion for Abstract Polymer Models. New Bounds from an Old Approach
We revisit the classical approach to cluster expansions, based on tree graphs, and establish a new convergence condition that improves those by Kotecký-Preiss and Dobrushin, as we show in some…
Statistical Mechanics: Rigorous Results
Thermodynamic behaviour - ensembles the thermodynamic limit for thermodynamic functions - lattice systems the thermodynamic limit for thermodynamic functions - continuous systems low density…
A phase cell cluster expansion for Euclidean field theories
Trees, forests and jungles: a botanical garden for cluster expansions
Combinatoric formulas for cluster expansions have been improved many times over the years. Here we develop some new combinatoric proofs and extensions of the tree formulas of Brydges and Kennedy, and…
...
...