Fluctuations of polymerized, fluid and hexatic membranes: continuum models and simulations

The thermal behavior of membranes - surfaces of ultra-low 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. Current research focuses on three areas: first, the properties of polymerized membranes with self-avoidance, attractive interactions, disorder, or external compression forces; second, the phase diagram and scaling behavior of fluid vesicles as a function of bending rigidity and pressure increment; and third, the destabilization of the crystalline phase by free dislocations at any finite temperature, and the properties of the resulting hexatic phase. Recent progress in this field is due to a fruitful interaction between theory and simulation studies of simple network models of membranes.