Geometrical Exponents of Contour Loops on Random Gaussian Surfaces

We derive the universal geometrical exponents of contour loops on equilibrium rough surfaces, using analytical scaling arguments (confirmed numerically): the fractal dimension $D_f$, the distribution of contour lengths, and the probability that two points are connected by a contour. This is sufficient to calculate exact critical exponents in certain nontrivial two-dimensional spin models that can be mapped to interface models. The novel scaling relation between $D_f$ and the roughness exponent that we find can be used to analyze scanning tunneling microscopy images of rough metal surfaces.