# Cluster expansions and correlation functions

@article{Ueltschi2003ClusterEA, title={Cluster expansions and correlation functions}, author={Daniel Ueltschi}, journal={arXiv: Mathematical Physics}, year={2003} }

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.

## 71 Citations

Abstract cluster expansion with applications to statistical mechanical systems

- Mathematics
- 2008

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

- Physics
- 2007

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

- Mathematics
- 2021

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

- Physics
- 2011

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

- Physics, Mathematics
- 2007

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

- Physics
- 2015

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

- Mathematics
- 2007

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

- Physics
- 2006

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

- Physics, MathematicsJournal of Mathematical Physics
- 2022

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

- MathematicsAnnales Henri Poincaré
- 2019

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…

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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…