Ballistic deposition on deterministic fractals: observation of discrete scale invariance

The growth of ballistic aggregates on deterministic fractal substrates is studied by means of numerical simulations. First, we attempt the description of the evolving interface of the aggregates by applying the well-established Family-Vicsek dynamic scaling approach. Systematic deviations from that standard scaling law are observed, suggesting that significant scaling corrections have to be introduced in order to achieve a more accurate understanding of the behavior of the interface. Subsequently, we study the internal structure of the growing aggregates that can be rationalized in terms of the scaling behavior of frozen trees, i.e., structures inhibited for further growth, lying below the growing interface. It is shown that the rms height (h_{s}) and width (w_{s}) of the trees of size s obey power laws of the form h_{s} proportional, variants;{nu_{ parallel}} and w_{s} proportional, variants;{nu_{ perpendicular}} , respectively. Also, the tree-size distribution (n_{s}) behaves according to n_{s} approximately s;{-tau} . Here, nu_{ parallel} and nu_{ perpendicular} are the correlation length exponents in the directions parallel and perpendicular to the interface, respectively. Also, tau is a critical exponent. However, due to the interplay between the discrete scale invariance of the underlying fractal substrates and the dynamics of the growing process, all these power laws are modulated by logarithmic periodic oscillations. The fundamental scaling ratios, characteristic of these oscillations, can be linked to the (spatial) fundamental scaling ratio of the underlying fractal by means of relationships involving critical exponents. We argue that the interplay between the spatial discrete scale invariance of the fractal substrate and the dynamics of the physical process occurring in those media is a quite general phenomenon that leads to the observation of logarithmic-periodic modulations of physical observables.