Capillary height-difference correlation ($B(r)=2(k_{B}T/\sigma )\int d^{D}q(2\pi {)}^{-D}(1-\cos (\mathbf{q}\cdot \mathbf{r}))/q^2$)

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

Adding positive quadratic confinement $t\int h^2/2$ to the membrane energy changes its height spectrum to $k_{B}T/(\sigma q^2+t)$. The continuous height-shift symmetry is explicitly broken. Connected correlation function decay on the finite length $\xi$ and the height-difference variance saturates for every internal dimension, although its saturation value can depend on a microscopic cutoff. A negative confinement coefficient gives an unstable quadratic model rather than pinning.