Truncated pair functions for free random surface models and Bernoulli ensembles are examined. In both cases, the pair function is shown to obey Ornstein-Zernike scaling whenever various correlation lengths of the system satisfy a nonperturbative criterion. Under the same conditions, the transverse displacement of surfaces contributing to the pair function is shown to be normally distributed. A new type of transition, which concerns the width of typical surfaces, is introduced and studied. Whenever the system is below the melting transition temperature of a related lower-dimensional model, the width of typical surfaces is shown to be finite. A thermodynamic formalism for free random surface models is developed. The formalism is used to obtain sharp estimates of the entropy of surfaces contributing to the pair function.