Unveiling the connection between the global roughness exponent and interface fractal dimension in EW and KPZ lattice models

A connection between the global roughness exponent and the fractal dimension of a rough interface, whose dynamics are expected to be described by stochastic continuum models, still needs more support from simulations in lattice models, which are key to provide completeness for the characterization of a given universality class. In this work, we investigate the asymptotic fractal dimension of interfaces that evolve according to some specific lattice models in d + 1 dimensions (d = 1, 2), which are expected to belong to the Edwards-Wilkinson or Kardar-Parisi-Zhang universality classes. Our results, based on the Higuchi method and on the extrapolation of the effective fractal dimension, allow one to achieve dependence between the asymptotic fractal dimension and global roughness exponent, in which the latter is expected to be hardly accessible for experimentalists. Conversely, we also use a two-points correlation function, which gives the time evolution of the local roughness exponent. As a byproduct, our results suggest that, for d = 1, the fractal dimension converges faster than the global roughness exponents to the asymptotic ones. Therefore, the analysis of the fractal dimension, for d = 1, is suggested to be more accessible than the global roughness exponents to determine the universality class. Corrections for the fractal dimensions in d = 2 were found to be stronger than for d = 1.