Discrete growth models on deterministic fractal substrate

The growth of the modified Family model and the Etching model on the Sierpinski carpet is studied by means of numerical simulations. The evolving interface of the aggregates is described by the well-established Family-Vicsek dynamic scaling approach. The results of the modified Family model prove the universality of the fractional Langevin equation introduced by Lee and Kim [S.B. Lee, J.M. Kim, Phys. Rev. E 80 (2009) 021101]. The Etching model also shows good scaling behavior. We conjecture that the systematic deviations of the data found in the ballistic deposition [C.M. Horowitz, F. Romá, E.V. Albano, Phys. Rev. E 78 (2008) 061118] may be due to the finite-size effects of the Ballistic Deposition model.