Cluster Expansions and Correlation Inequalities

Cluster and polymer expansions turn a many-body exponential into a sum of connected contributions whose absolute convergence is controlled by a quantitative smallness condition. Within that domain they prove analyticity and decay of truncated correlations. Outside it, the method is silent: the model may still exist, but analytic continuation of the expansion is not a proof.

Required background. Interacting measures, stability, and Wick ordering supplies the normalized finite-volume law; clustering, vacuum uniqueness, and mass-gap implications supplies the meaning of connected decay.

Helpful background. Gaussian Euclidean fields as measures supplies covariance decay; the constructive program and cutoff removal explains why the bounds must be uniform.

Connected polymers and a convergence criterion

Partition space into unit blocks and write a regulated interaction as a product of local factors. Expanding those factors and integrating the Gaussian field groups mutually linked blocks into polymers $X$. The partition function has the hard-core form

$$Z_\Lambda=Z_{0,\Lambda}\sum_{\{X_1,\ldots,X_n\}\,\mathrm{compatible}} \prod_{i=1}^n z(X_i),$$

where compatibility means disjointness and $z(X)$ is a polymer activity. A Kotecký–Preiss-type hypothesis is: there is a size function $a(X)\ge0$ such that

$$\sum_{Y:Y\not\sim X}|z(Y)|e^{a(Y)}\le a(X)$$

for every $X$. Then the logarithm is an absolutely convergent sum over connected incompatibility graphs. Differentiating source-dependent activities gives absolutely convergent expansions for truncated correlations. Exponential weights in $a(X)$ transfer to exponential decay in the separation of source supports.

The theorem’s domain includes a named block decomposition, activities, compatibility relation, and norm. Its conclusion is absolute convergence and the consequences obtained by legitimate termwise operations. The converse is false: failure of this sufficient inequality neither proves a phase transition nor disproves existence.

Weakly coupled massive P(φ)₂

For massive $P(\phi)_2$ with $P$ bounded below, the covariance between separated blocks decays exponentially. At sufficiently small dimensionless coupling $\lambda/m_0^2$, local large-field estimates control the activities while the mass controls the links. The expansion for connected Schwinger functions converges uniformly in growing rectangles and yields

$$|S^T_{p+q}(f_1,\ldots,f_p,g_{1,a},\ldots,g_{q,a})| \le C_{f,g}e^{-m|a|}$$

for separated test-function families. Glimm, Jaffe, and Spencer used precisely this route to obtain the infinite-volume $P(\phi)_2$ Schwinger functions, OS axioms, exponential clustering, and an isolated particle mass; Summers 2016, §3.1, pp. 11–12 summarizes the hypotheses and conclusions.

The connected expansion is checked combinatorially by setting all activities except single-block activities to zero: then $\log Z$ is a sum of independent block contributions and every connected correlation between distinct blocks vanishes. At first nontrivial order, only connected graphs joining all marked supports may contribute. A disconnected graph appearing in a truncated function signals a missing cumulant subtraction.

This construction informs strong-coupling phases and cross-method evidence by supplying rigorous weak-coupling control. It does not transfer its analyticity domain into the strong-coupling regime.

Correlation inequalities are a different route

For ferromagnetic even interactions, positivity inequalities can give monotonicity in the volume and bounds without requiring small coupling. They exploit an order structure absent from a generic polymer expansion. In $P(\phi)_2$, the GKS inequalities imply nonnegative correlations and supermultiplicative moment inequalities; together with suitable boundary conditions they support a thermodynamic limit. Summers 2016, §3.1, pp. 11–13 distinguishes this unrestricted-coupling but interaction-restricted route from weak-coupling cluster expansions.

Neither method contains the other. A convergent expansion can treat interactions without ferromagnetic correlation inequalities; an inequality can control parameters where the expansion norm is too large. Their conclusions must be read from their own hypotheses.

Adversarial failure test

Increase $\lambda$ until the estimated sum of overlapping activities exceeds $a(X)$. The graph series is then no longer absolutely controlled. Reusing its decay bound by “analytic continuation” assumes the very absence of singularities that a phase transition may violate. The strongest honest statement is that this expansion no longer decides the correlations.

Likewise, replacing a massive covariance by a massless one removes exponential block-to-block decay. A power-law weighted polymer norm might still work in a different theorem, but the massive exponential estimate cannot simply be retained.

Exercises

1. Two independent polymers. If polymers $X$ and $Y$ never conflict and $Z=(1+z_X)(1+z_Y)$, show that the mixed connected contribution vanishes.

Solution

$\log Z=\log(1+z_X)+\log(1+z_Y)$ has no mixed monomial. This is the finite example of the linked-cluster principle.

2. A weighted geometric bound. Suppose every polymer containing a fixed block has total activity of size $n$ bounded by $q^n$ and $q e^\alpha<1$. Show the exponentially weighted sum converges.

Solution

It is bounded by $\sum_{n\ge1}(qe^\alpha)^n=qe^\alpha/(1-qe^\alpha)$. The strict inequality is the smallness margin that can pay for graph combinatorics.

References

• Brydges, David C. “A Short Course on Cluster Expansions.” In Critical Phenomena, Random Systems, Gauge Theories, 129–183. Les Houches 1984. North-Holland, 1986. Open PDF.
• Glimm, James, Arthur Jaffe, and Thomas Spencer. “The Wightman Axioms and Particle Structure in the $P(\phi)_2$ Quantum Field Model.” Annals of Mathematics 100 (1974): 585–632. DOI.
• Summers, Stephen J. “A Perspective on Constructive Quantum Field Theory.” arXiv:1203.3991, revised 2016. Open PDF.