= Capillary height-difference correlation
{title2=$B(r)=2(k_BT/\sigma)\int d^Dq\,(2\pi)^{-D}(1-\cos(\mathbf q\cdot\mathbf r))/q^2$}
For a tension-dominated membrane with internal dimension $D$, the massless Gaussian height <covariance> has spectrum $k_BT/(\sigma q^2)$. Removing the uniform translation mode leaves the height difference well-defined. Its <variance> grows linearly with distance for $D=1$, logarithmically for $D=2$, and approaches a cutoff-dependent constant for $D>2$. A microscopic cutoff is necessary when the short-distance integral diverges. This distinguishes rough positional order from a failure of the local small-slope approximation.
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