Cluster expansion theorem (analyticity from convergent polymer expansions)

Abstract conditions ensuring convergence of cluster expansions for log-partition functions, yielding analyticity of pressure and decay of correlations at high temperature or low density.

Cluster expansion theorem (analyticity from convergent polymer expansions)

Context

Cluster expansions turn a partition function into a controlled series over connected objects (polymers/contours), yielding:

analyticity of the pressure (log-partition density) ,
uniqueness of the Gibbs state in appropriate regimes,
quantitative bounds on correlations and a finite correlation length .

In lattice spin systems these regimes typically correspond to high temperature (small $\beta$ ) or, in gas models, low activity/density.

Definition (abstract polymer model)

Let $\mathcal{P}$ be a set of polymers with:

a compatibility relation: $\gamma\sim\gamma'$ means polymers are compatible (can coexist),
an activity (weight) $z(\gamma)\in\mathbb{C}$ .

For a finite region $V$ , the polymer partition function is

Z_V = \sum_{\Gamma\ \text{compatible in }V}\ \prod_{\gamma\in\Gamma} z(\gamma),

where the sum runs over finite compatible families $\Gamma$ (including the empty family).

The associated finite-volume pressure is

p_V = \frac{1}{|V|}\log Z_V,

which matches the usual canonical setup (see partition function ).

Theorem (cluster expansion; typical Kotecký–Preiss-type criterion)

Assume there exists a function $a:\mathcal{P}\to(0,\infty)$ such that for every polymer $\gamma$ ,

\sum_{\gamma'\not\sim \gamma} |z(\gamma')|\,e^{a(\gamma')} \le a(\gamma).

Then:

Convergent expansion for $\log Z_V$ :
$\log Z_V = \sum_{n\ge 1}\ \frac{1}{n!}\sum_{(\gamma_1,\dots,\gamma_n)} \phi(\gamma_1,\dots,\gamma_n)\prod_{i=1}^n z(\gamma_i),$
where $\phi$ vanishes unless the incompatibility graph on $\{\gamma_1,\dots,\gamma_n\}$ is connected (Ursell/connected cluster coefficients).
Absolute convergence and analyticity: the series converges absolutely and uniformly in $V$ in a polydisc of activities, implying that $p_V$ and the infinite-volume pressure are analytic functions of the parameters in that regime.
Uniqueness and decay of correlations (in applications): when the polymer representation encodes a Gibbs measure, convergence typically implies uniqueness of the infinite-volume Gibbs state and exponential decay of truncated correlations, hence a finite correlation length.

Typical applications in statistical mechanics

High-temperature spin systems: expansions around $\beta=0$ prove analyticity and uniqueness (useful for high-temperature CLT results and mixing estimates).
Low-density gases: the expansion yields virial/cluster integral relations (compare Mayer cluster integrals and virial coefficients ) and convergence criteria (see virial expansion convergence ).
Contours at low temperature: contour expansions are a key input to Pirogov–Sinai theory for phase coexistence and surface tension.

Prerequisites and connections (cross-links)

Canonical thermodynamics link: canonical ensemble , free energy , pressure as log-partition density .
Gibbs formalism: finite-volume Gibbs measures , DLR equations .
Phase transitions: phase transitions (Gibbs) , equivalent indicators of phase transitions .