Exotic marginal fractals from hierarchical random deposition

We study marginal forms on the borderline between Euclidean shapes and fractals. The fractal dimension equals the topological dimension. The fractal measure features a logarithmic correction factor, leading to a linear divergence of the area or length upon increasing the resolution instead of the usual exponential law for fractals. We discuss a physical model of random deposition of particles or fragments obeying a strict size hierarchy. The landscapes resulting from this deposition process resemble modern cities and satisfy the logarithmic fractal law. The roughness of the shapes is characterized by a fractal amplitude, which can be calculated exactly as a function of the (time-dependent) probabilities and rescaling factor of the random deposition process. Besides possible realizations in particle or cluster deposition physics and colloid physical chemistry, an application to the science of porous media in the plane is described.