Network models of fluid, hexatic and polymerized membranes

The thermal behaviour of membranes - surfaces of nearly vanishing tension - depends strongly on their internal state, which can be either fluid, crystalline (or polymerized), or hexatic. Thermal fluctuations have a dramatic effect on the conformation and elastic properties of membranes. We describe in this review both the continuum models of membranes which are used for theoretical analyses as well as the network models employed in simulations. The fruitful interaction between these two approaches, which has lead to recent progress in this field, is emphasized. We summarize the essential results of recent research on fluctuating membranes; in particular, the effects of bending rigidity, self-avoidance, attractive interactions, disorder, topological defects and external compression forces are discussed in detail.