Central Limit Theorem in the High-Temperature Gibbs Regime

A CLT for spatial sums of local observables under a unique high-temperature Gibbs measure, with variance given by the integrated covariance (susceptibility-type) formula.

Central Limit Theorem in the High-Temperature Gibbs Regime

Prerequisites

Setting

Let $\mu$ be a translation-invariant infinite-volume Gibbs measure on $\mathbb{Z}^d$ (see infinite-volume Gibbs measures ), arising from a finite-range interaction in a high-temperature / uniqueness regime (for example, a parameter region where a cluster expansion yields exponential decay of correlations).

Let $f$ be a local observable (depends only on finitely many spins) with zero mean:

\mu(f)=0.

Let $\tau_x$ denote the lattice shift by $x$ , and define the sum over a finite region $\Lambda\subset\mathbb{Z}^d$ :

S_\Lambda = \sum_{x\in\Lambda} f\circ \tau_x.

Theorem (CLT under high-temperature mixing)

Assume $\mu$ is strongly mixing with summable covariances for $f$ , i.e.

\sum_{x\in\mathbb{Z}^d} \big|\mathrm{Cov}_\mu\big(f, f\circ\tau_x\big)\big| < \infty.

Then, for a sequence of boxes $\Lambda_n$ with $|\Lambda_n|\to\infty$ ,

\frac{S_{\Lambda_n}}{\sqrt{|\Lambda_n|}} \;\Rightarrow\; \mathcal{N}(0,\sigma_f^2),

where the asymptotic variance is given by the integrated covariance formula

\sigma_f^2 ={} \sum_{x\in\mathbb{Z}^d} \mathrm{Cov}_\mu\big(f, f\circ\tau_x\big) \;\;\in\;[0,\infty).

Moreover,

\lim_{n\to\infty}\frac{\mathrm{Var}_\mu(S_{\Lambda_n})}{|\Lambda_n|}=\sigma_f^2.

A multivariate version holds for finitely many local observables $(f_1,\dots,f_k)$ , yielding a Gaussian limit with covariance matrix given by the corresponding summed cross-covariances.

Interpretation in statistical mechanics

The high-temperature (uniqueness) regime typically implies exponential decay of correlations and a finite correlation length , which is precisely what makes the covariance sum converge.
The variance formula above is the $k=0$ limit of a Fourier-space fluctuation observable; it connects naturally to structure factor .
At or near criticality (large correlation length), the covariance sum may diverge or decay too slowly, and the CLT can fail or require non-Gaussian scaling limits (see critical exponents and scaling relations for how this is encoded in critical behavior).