Convergence of the cluster expansion

Statement (cluster expansion convergence)

Many lattice and continuum models can be rewritten as a polymer model whose (finite-volume) partition function has the form

where:

• $\Gamma$ ranges over finite collections of mutually compatible polymers $\gamma$ (no hard-core conflicts),
• $w(\gamma)$ are (possibly complex) polymer weights/activities.

Assume there exists a nonnegative function $a(\gamma)\ge 0$ such that the Kotecký–Preiss (KP) criterion holds:

where $\gamma' \not\sim \gamma$ means $\gamma'$ is incompatible with $\gamma$.

Then:

1. The cluster expansion for $\log Z$ converges absolutely:

   $$\log Z = \sum_{n\ge 1} \frac{1}{n!}\sum_{\gamma_1,\dots,\gamma_n} \phi^T(\gamma_1,\dots,\gamma_n)\prod_{i=1}^n w(\gamma_i),$$

   where $\phi^T$ is the usual truncated/connected coefficient (Ursell function; a signed sum over connected graphs).

2. The expansion converges uniformly in volume whenever the KP bound is uniform, yielding existence of the infinite-volume pressure and analyticity in the parameters appearing in $w(\gamma)$.

3. In applications where correlation functions can be expanded similarly, the same convergence mechanism yields strong control of truncated correlations and, under standard geometric assumptions, exponential decay of correlations.

Key hypotheses

• A polymer representation of the relevant finite-volume partition function (often obtained from a high-temperature expansion, Mayer expansion, or contour representation).
• Absolute summability/smallness encoded by the KP criterion: $\sum_{\gamma' \not\sim \gamma} |w(\gamma')|e^{a(\gamma')}\le a(\gamma).$
• (For thermodynamic-limit statements) uniformity of the KP constants with respect to the volume and boundary conditions.

Key conclusions

• Absolute convergence of $\log Z$ as a sum over connected clusters.
• Analyticity of $\log Z$ (hence of the finite-volume pressure) in the parameters controlling $w(\gamma)$, within the KP domain.
• Under standard additional bookkeeping, existence of the thermodynamic limit of the pressure (often as in thermodynamic-limit pressure existence ) and exponential decay of truncated correlations.

Cross-links to relevant definitions

• Finite-volume partition functions in lattice systems: partition function (lattice) .
• Finite-volume and infinite-volume pressure: pressure (lattice) .
• Gibbs measures used to interpret expectations/correlations: finite-volume Gibbs measure and infinite-volume Gibbs measure .
• Correlations whose decay is often derived from convergent expansions: two-point correlation function .
• Grand-canonical viewpoint (when polymers arise from activity/fugacity expansions): grand canonical ensemble .

Proof idea / significance (sketch)

The cluster expansion is obtained by writing $\log Z$ as a sum of connected contributions (via inclusion–exclusion or a graph expansion). The KP criterion provides a dominating tree bound: connected graph sums are controlled by spanning trees, and the KP inequality forces the resulting series to be absolutely summable. Uniform convergence in volume then implies analyticity of the pressure and, when applied to truncated correlation expansions, yields quantitative decay estimates.