Slow decay of the direct correlation function in the three-dimensional capillary-wave model of the interface

The explicit expression in two and three dimensions for the conditional direct correlation function $C_{\mathrm{cond}}$, representing the intrinsic part of the direct correlation function C, is obtained in the capillary-wave model in the limit of unbounded interface fluctuations. $C_{\mathrm{cond}}$ varies in the transverse and the longitudinal directions on intrinsic interface scales, which are expressed only in terms of the bulk correlation length ${\xi }_b$, the surface tension σ, and the temperature T, thus are independent of external conditions stabilizing the interface in space. For d=2, $C_{\mathrm{cond}}$ is strictly short ranged, whereas for d=3, $C_{\mathrm{cond}}$ is long ranged and its Fourier transform in the transverse direction (parallel to the interface) is not analytic at $k_{\perp }$=0 in contrast to the usual assumptions. The relation to the Yvon-Triezenberg-Zwanzig formula is clarified.