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Each nonzero mode is Gaussian, giving the covarianceConsequently the capillary height-difference correlation isThe difference removes the zero mode and makes the integral infrared-finite at each fixed separation. Where required, cut off $q$ at $\Lambda \sim 1/a$ with microscopic length $a$. The question's $d$ is the embedding dimension, so the momentum integral has dimension $D=d-1$, not $d$.

For $d=2$, $D=1$, use ${\int }_{-\infty }^{\infty }dq(1-\cos qr)/(2\pi q^2)=|r|/2$. HenceThe root-mean-square relative height grows as $r^{1/2}$: the interface wanders increasingly far from a fixed height.

For $d=3$, $D=2$, angular integration givesThe Bessel function expression separates the universal logarithmic growth from a cutoff-dependent constant. The membrane is logarithmically rough, so its absolute height fluctuations are unbounded in the thermodynamic limit even though long-wavelength slopes need not be large.

For $d=4$, $D=3$, the massless Green function at nonzero separation is $1/(4\pi r)$. Thuswhere $B_{\infty }$ is finite once the microscopic cutoff is supplied. For a spherical sharp cutoff, $B_{\infty }=k_{B}T\Lambda /({\pi }^2\sigma )$. Relative height fluctuations saturate at large distance, allowing a localized interface height selected by boundaries. The roughness distinction is linear growth in $d=2$, logarithmic growth in $d=3$, and bounded long-distance fluctuations in $d=4$. The short-distance cutoff and the small-slope assumption remain necessary; bounded height fluctuations do not imply a perfectly flat microscopic membrane.