Home » Classical Statistical Mechanics

Convergence of the cluster expansion

Sufficient small-activity/high-temperature conditions guaranteeing absolute convergence of the cluster expansion for log partition functions, with analyticity of the pressure and control of correlations.
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

Z=∑Γ compatible ∏γ∈Γw(γ), Z = \sum_{\Gamma \ \text{compatible}} \ \prod_{\gamma \in \Gamma} w(\gamma),

where:

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

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

∑γ′≁γ∣w(γ′)∣ ea(γ′)≤a(γ)for all polymers γ, \sum_{\gamma' \not\sim \gamma} |w(\gamma')|\,e^{a(\gamma')} \le a(\gamma)\qquad \text{for all polymers }\gamma,

where γ′≁γ\gamma' \not\sim \gamma means γ′\gamma' is incompatible with γ\gamma.

Then:

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

    log⁡Z=∑n≥11n!∑γ1,…,γnϕT(γ1,…,γn)∏i=1nw(γi), \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 ϕT\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(γ)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: ∑γ′≁γ∣w(γ′)∣ea(γ′)≤a(γ). \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\log Z as a sum over connected clusters.
  • Analyticity of log⁡Z\log Z (hence of the finite-volume pressure) in the parameters controlling w(γ)w(\gamma), within the KP domain.
  • Under standard additional bookkeeping, existence of the thermodynamic limit of the pressure (often as in ) and exponential decay of truncated correlations.