Cluster expansions and correlation functions

@article{Ueltschi2003ClusterEA,
  title={Cluster expansions and correlation functions},
  author={Daniel Ueltschi},
  journal={arXiv: Mathematical Physics},
  year={2003}
}
  • D. Ueltschi
  • Published 1 April 2003
  • Mathematics
  • arXiv: Mathematical Physics
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 estimates for correlation functions are also presented. The results are applied to systems of interacting classical and quantum particles, and to a lattice polymer model. 

Figures from this paper

Abstract cluster expansion with applications to statistical mechanical systems
TLDR
A general setting for the cluster expansion method is formulated and sufficient criteria for its convergence is discussed, and the results are applied to systems of classical and quantum particles with stable interactions.
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
Cluster expansions: Necessary and sufficient convergence conditions
We prove a new convergence condition for the activity expansion of correlation functions in equilibrium statistical mechanics with possibly negative pair potentials. For non-negative pair potentials,
A cluster expansion approach to renormalization group transformations
The renormalization group (RG) approach is largely responsible for the considerable success which has been achieved in developing a quantitative theory of phase transitions. This work treats the
The model of interacting spatial permutations and its relation to the Bose gas
The model of spatial permutations is related to the Feynman-Kac representation of the Bose gas. The transition to infinite cycles corresponds to Bose-Einstein condensation. We review the general
A Cluster Expansion Approach to the Heilmann–Lieb Liquid Crystal Model
A monomer-dimer model with a short-range attractive interaction favoring collinear dimers is considered on the lattice $$\mathbb Z^2$$Z2. Although our choice of the chemical potentials results in
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
Mott Transition in Lattice Boson Models
We use mathematically rigorous perturbation theory to study the transition between the Mott insulator and the conjectured Bose-Einstein condensate in a hard-core Bose-Hubbard model. The critical line
Cluster expansion for a dilute hard sphere gas dynamics
In a previous paper by the authors, a cluster expansion method had been developed to study the fluctuations of the hard sphere dynamics around the Boltzmann equation. This method provides a precise
Cluster Expansions with Renormalized Activities and Applications to Colloids
We consider a binary system of small and large objects in the continuous space interacting via a nonnegative potential. By integrating over the small objects, the effective interaction between the
...
...

References

SHOWING 1-10 OF 15 REFERENCES
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.”
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
Decay of correlations for infinite range interactions in unbounded spin systems
In unbounded spin systems at high temperature with two-body potential we prove, using the associated polymer model, that the two-point truncated correlation function decays exponentially
General properties of polymer systems
We prove the existence of the thermodynamic limit for the pressure and show that the limit is a convex, continuous function of the chemical potential.The existence and analyticity properties of the
A Gentle Introduction to Cluster Expansions
A measure on a high dimensional space is to be defined by a density with respect to a reference measure. The goal is to control this measure in the limit as the dimension becomes infinite. A cluster
A gentle introduction to cluster expansions
polymer system: t(p, q) = 1 or 0. Also t(p, p) = 1. Incompatible sites: I(p) = {q | t(p, q) = 1}. Tp(M) = 1− (1− tp) is 1 or 0 if M(I(p)) > 0 or = 0. Interaction bound lemma: Tp(M) ≤ M(I(p)). Second
...
...