Nonuniversal diffusivity exponent for the soft-percolation process in two dimensions.

Abstract

Soft percolation is a dynamic percolation process in the spatially disordered system, where the jump rate between two sites is assumed to have power-law dependence on the distance between the sites with a cutoff. We present a numerical study of the soft percolation model in two dimensions. We first derive an exact expression for the diffusion constant of a system with periodic boundary conditions. We obtain the diffusion constant as a function of the density of sites for various samples generated by a Monte Carlo technique and find, using the finite size scaling, that for 01, \ensuremath{\mu} is almost equal to the known value ${\mathrm{\ensuremath{\mu}}}_{\mathit{u}\mathit{n}}$\ensuremath{\approxeq}1.30 for the ordinary percolation process and for \ensuremath{\alpha}>1, \ensuremath{\mu} increases linearly in \ensuremath{\alpha}, where \ensuremath{\alpha} denotes the exponent of the power-law dependence. This result indicates that the contrauniversality holds, that is, the critical percolation density is independent of \ensuremath{\alpha} and the diffusivity exponent \ensuremath{\mu} depends on \ensuremath{\alpha}.