Statistical Mechanics of Tethered Surfaces

We study the statistical mechanics of two-dimensional surfaces of fixed connectivity embedded in $d$ dimensions, as exemplified by hard spheres tethered together by strings into a triangular net. Without self-avoidance, entropy generates elastic interactions at large distances, and the radius of gyration $R_G$ increases as ${\left(\ln L\right)}^{\frac{1}{2}}$, where $L$ is the linear size of the uncrumpled surface. With self-avoidance $R_G$ grows as $L^{\nu }$, with $\nu =\frac{4}{\mathrm{}(d+2\mathrm{})\mathrm{}}$ as obtained from a Flory theory and in good agreement with our Monte Carlo results for $d=3\mathrm{}$.