Ensemble of Interacting Random Surfaces On A Lattice

We study an ensemble of interacting closed random surfaces on a cubic lattice. The statistical weight of each surface configuration depends on the total surface area, on the mean (or extrinsic) curvature and on an interaction term arising when two surfaces touch each other along some contour. Part of this interaction is shown to have an entropic origin. The equivalence with a spin model is used to determine the exact phase diagram at zero temperature. At T not-equal 0 some regions of the phase diagram are investigated by local mean field approximations and Monte Carlo simulations. We find homogeneous phases, disordered and ordered (cubic) bicontinuous phases, and two droplet crystal phases differing by the presence of a spontaneous curvature in one of them. All these phases appear without introducing any intrinsic curvature energy. The model is generalized in order to include surfaces with spontaneous curvature or open surfaces whose boundaries are weighted.