Statistical properties of random fractals : geometry, growth and interface dynamics

Abstract

This thesis comprises analytic and numerical studies of static, geometrical properties of fractals and dynamical processes in them. First, we have numerically estimated the subset fractal dimensions DS describing the scaling of some subsets S of the fractal cluster with the linear cluster size R in the q-state Potts models. These subsets include the total mass of the cluster, the hull, the external perimeter, the singly connected bonds and the gates to fjords. Numerical data reveals complex corrections-to-scaling behavior needed to take into account for correct extrapolation of the data to the asymptotic large size limit. Using renormalization group theory the corrections-to-scaling terms are analytically derived. The numerical data are in good agreement with exact values of the fractal dimensions and with the exactly predicted correction terms. Regarding the growth of fractal structures, we consider 2D continuum deposition models which generate fractal structures from the point of view of percolation theory. In the particular model studied here, there is an effective inter-particle rejection. Using previous results from related irreversible deposition models mean field predictions for the percolation thresholds of the model are derived in the limits of the parameter space defining the model. Numerical simulations of the model support the theoretical results. The networks exhibit non-trivial spatial correlations, which manifest themselves in a power-law behavior of the mass density fluctuation correlation function for small distances. Geometric properties of fractals play a crucial role in the dynamics and kinetic roughening of driven fronts in fractals. Here we show that for the isotropic invasion percolation model, an algebraically decaying distribution for the nearest neighbor slope distribution of single-valued fronts follows from scaling arguments derived using the properties of percolation clusters. From the distribution, the form of which is also valid for anisotropic cases such as the diffusion limited aggregation model, the exponents governing the scaling of various spatio-temporal correlation functions are derived. The results indicate that the fractal growth models exhibit intrinsic anomalous scaling and multiscaling. Numerical simulations show excellent agreement with the predictions.