Cluster expansion (construction)

Write log of the partition function as a convergent sum over connected clusters (polymers/graphs), typically in a high-temperature or low-activity regime.

Cluster expansion (construction)

A cluster expansion is a systematic way to rewrite the logarithm of a finite-volume partition function as a sum of connected contributions. It is the workhorse behind rigorous high-temperature/low-density results: analyticity of the free energy, existence of the thermodynamic limit , and decay of correlations.

Polymer representation

In many lattice and continuum settings (e.g. models on a finite box inside a $\mathbb{Z}^d$ lattice ), one first rewrites the finite-volume partition function—such as the canonical partition function or the grand partition function —in polymer-gas form:

Z_\Lambda \;=\; \sum_{\Gamma \,\text{compatible}} \;\prod_{\gamma\in\Gamma} w(\gamma).

Here:

$\Lambda$ is a finite region (volume).
$\mathcal P_\Lambda$ is a collection of “polymers” $\gamma$ (typically localized objects: contours, connected sets, interaction blocks).
$w(\gamma)$ is the polymer activity (weight).
A set $\Gamma\subset \mathcal P_\Lambda$ is compatible if its polymers do not overlap (or otherwise satisfy a model-dependent compatibility rule).

This representation is the starting point for the general construction of partition functions in regimes where interactions can be treated perturbatively.

Cluster expansion for $\log Z_\Lambda$

The cluster expansion is the identity

\log Z_\Lambda \;=\; \sum_{n\ge 1}\frac{1}{n!} \sum_{(\gamma_1,\dots,\gamma_n)\in\mathcal P_\Lambda^n} \phi^T(\gamma_1,\dots,\gamma_n)\; \prod_{i=1}^n w(\gamma_i),

where $\phi^T(\gamma_1,\dots,\gamma_n)$ is the truncated (connected) coefficient. For a polymer gas with a hard incompatibility relation, define

f(\gamma_i,\gamma_j)= \begin{cases} -1, & \gamma_i \text{ incompatible with } \gamma_j,\\ 0, & \gamma_i \text{ compatible with } \gamma_j. \end{cases}

Then one convenient formula is

\phi^T(\gamma_1,\dots,\gamma_n) \;=\; \sum_{\substack{G\ \text{connected graph}\\\text{on }\{1,\dots,n\}}} \ \prod_{(i,j)\in E(G)} f(\gamma_i,\gamma_j).

Only connected graphs contribute, which is why $\log Z_\Lambda$ is a sum over connected “clusters.”

This “connected vs disconnected” structure is the same reason derivatives of $\log Z$ produce cumulants and connected correlators (see connected correlations from cumulants and fluctuation formulas from $\log Z$ ).

Convergence and consequences

A cluster expansion becomes useful when the right-hand side converges absolutely and uniformly in $\Lambda$ . A standard sufficient condition is of Kotecký–Preiss/Dobrushin type: there exists a size function $a(\gamma)\ge 0$ such that for every polymer $\gamma$ ,

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

where $\gamma'\not\sim \gamma$ means “incompatible.”

When such a condition holds:

$(1/|\Lambda|)\log Z_\Lambda$ has a limit as $|\Lambda|\to\infty$ , giving the free energy density via free energy from the partition function and the pressure via pressure from the partition function .
Connected correlations decay rapidly, implying a finite correlation length in that regime.
One can often construct the infinite-volume Gibbs state by taking a weak limit of finite-volume Gibbs measures .

Physical interpretation

The expansion organizes thermodynamics by interaction clusters: a cluster term represents a small group of degrees of freedom whose interactions cannot be factorized into independent pieces. Disconnected collections factorize in $Z_\Lambda$ but cancel in $\log Z_\Lambda$ , leaving only genuinely collective contributions.

A canonical continuum example of the same connected-graph mechanism is the Mayer expansion for gases.

Polymer representation

Cluster expansion for log⁡ZΛ\log Z_\LambdalogZΛ​

Convergence and consequences

Physical interpretation

Cluster expansion for $\log Z_\Lambda$