Edge correlations of fluid and tethered membranes

The fluctuations of fluid and polymerized open membranes near a free edge are studied analytically, and using Monte Carlo simulations and scaling arguments. Its is shown that flat, finite (or semi-infinite) fluid membranes with zero edge energy (line tension) and zero Gaussian curvature energy are unstable with respect to non-planar deformations, even on length scales small compared to the persistence length. The flat phase can, however, be stabilized by either a line tension $\delta$, or by a Gaussian curvature with a saddle-splay modulus $\bar\kappa$ in the range $-4\kappa < \bar \kappa < 0$. At the mean-field level, we find $\zeta_{\parallel}-\zeta_{\perp}=\zeta=1$ for stable membranes when $\sigma=0$, and $\zeta_{\parallel}=0.5$ for a finite line tension, where $\zeta_{\parallel}$ and $\zeta_{\perp}$ characterize the decay of correlations in the directions parallel and perpendicular to the membrane edge, and $\zeta$ is the bulk exponent. For tethered membranes, simulation results and scaling arguments imply $\zeta_{\parallel}=\zeta_{\perp}=\zeta$, with $\zeta =$ 0.70