The Statistical Properties of Random Surfaces

Available from UMI in association with The British Library. Requires signed TDF. The statistical properties of random surfaces are relevant in many otherwise disparate realms of modern science, ranging from the biological, to the chemical, to the physical. The current understanding of surface behaviour is strictly limited to the perturbative and mean field limits. Several early attempts to investigate the low-dimensional non-perturbative regime have painted fuzzy, and often contradictory, pictures. It is the purpose of this thesis to examine rigorously the non-perturbative behaviour of surfaces in three and four dimension, clearing up as many of the as yet unresolved problems as possible. To that end, two discrete models whose naive continuum limits lead to Polyakov's string field theory, in which the action includes terms proportional both to the area of the surface and to its extrinsic curvature, have been studied using a Fourier-accelerated Langevin simulation on lattices containing up to 128^2 sites. Both models are seen to undergo continuous phase transitions between 'crumpled' and 'smooth' phases; however, the evidence suggests that the models lie in different universality classes. The nature and location of the phase transitions have been accurately determined, and critical exponents related to the specific heat, the correlation length, and the extrinsic scaling properties of the surface have been measured. Both models have an infinite 'effective dimension' in the 'crumpled' phase, i.e. they are space filling in any number of embedding dimensions, but differ considerably in the 'smooth' phase due to the presence of an anisotropic low-action configuration which is a lattice artefact in the first model.