Crumpled and Flat Regimes In A Random Surface Model

A spherical like model of a D-dimensional random surface embedded in d-dimensional Euclidean space is studied in detail. The embedding weight depends on an attractive term between the nearest neighbours and on a repulsive one between some of the next to the nearest neighbours of the network. The repulsive term mimics an extrinsic curvature energy for surface configurations. Crumpled and flat regimes are found, and, if D less-than-or-equal-to 2, only the former survives in the thermodynamic limit. The model can be seen as the d --> infinity limit of a more realistic model where the 1/d corrections stabilize the flat regime in the thermodynamic limit at least for D = 2.