Random surfaces from hierarchical deposition of debris with alternating rescaling factors

Abstract

An analytical study of surface profiles that result from hierarchical random impact of debris on the line is performed in terms of logarithmic fractals theory. The hierarchical random deposition model is extended for the case of time-dependent probabilities P (for positioning a hill on the surface) and Q (for digging a hole) and spatial rescaling factor λ. The periodic deposition model is solved exactly, and the logarithmic fractal roughness of the surface profile is found to be robust with respect to time-dependent perturbations. The fractal amplitudes associated with the proliferation of the surface length are compared with those calculated in the static regime and are shown to have a nontrivial interaction. It is verified that amplitude repulsion, attraction, neutrality, and auto-repulsion take place. The transient regime is also studied and is shown to have exponential decay towards the asymptotic regime. Special attention is devoted to the case of alternating rescaling factors, for which new results are derived.