 Capillary height-difference correlation

For a tension-dominated membrane with internal dimension $D$, the massless Gaussian height covariance has spectrum $k_{B}T/(\sigma q^2)$. 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.