Finite-size scaling theory for anisotropic percolation models

Finite-size scaling (FSS) theory for anisotropic percolation models is rarely studied. A simple FSS theory is developed here for anisotropic percolation models considering the cluster size distribution function as a generalized homogeneous function of the system size L and two connectivity lengths xi(parallel to) and xi(perpendicular to). The scaling theory predicts a new FSS function form for the cluster related quantities in terms of the anisotropic exponent theta = v parallel to/v(perpendicular to), where v(parallel to) and v(perpendicular to) are the connectivity exponents in the longitudinal and transverse directions respectively and a set of new scaling relations are obtained. In the directed percolation (DP) and directed spiral percolation (DSP) models, the clusters generated are anisotropic and they are called anisotropic percolation models. The FSS theory developed here is verified applying to the DP and DSP models.