5 - Cluster expansions

Summary

Derrière la série de Fourier, d'autres séries analogues sont entrées dans la domaine de l'analyse; elles y sont entrées par la même porte; elles ont été imaginées en vue des applications.

Henri Poincaré, La valeur de la science.

Most computational methods in statistical mechanics rely upon perturbation theory around situations that are well understood. The simplest one is, as always, the ideal gas. Expansions around the ideal gas are known as high-temperature or weak-coupling expansions. The other type of expansions concern the situation when the Gibbs measure concentrates near a single ground-state configuration. Such expansions are known as low-temperature expansions. Technically, in both cases, they involve a reformulation of the model in terms of what is called a polymer model. We begin with the high-temperature case, which is both simpler and less model-dependent than the low-temperature case, and show how a polymer model is derived.

High-temperature expansions

We place ourselves in the context of regular interactions, and we assume that β will be small. In this situation, we can expect that our Gibbs measure should behave like a product measure. To analyze such a situation, we will always study the local specifications, establishing that they depend only weakly on boundary conditions.