Anisotropic bond percolation

Abstract

The authors introduce anisotropic bond percolation in which there exist different occupation probabilities for bonds placed in different coordinate directions. They study in detail a d-dimensional hypercubical lattice, with probabilities p_{perpendicular to} for bonds within (d-1)-dimensional layers perpendicular to the z direction, and p/sub /// identical to Rp_{perpendicular to} for bonds parallel to z. For this model, they calculate low-density series for the mean size S, in both two and three dimensions for arbitrary values of the anisotropy parameter R. It is found that in the limit 1/R to 0, the model exhibits crossover between 1 and d-dimensional critical behaviour, and that the mean-size function scales in 1/R. From both exact results and series analysis, the crossover exponent ( identical to phi ₁) is 1 for all d, and that the divergence of successive derivatives of S with respect to 1/R increases with a constant gap equal to 1 in two and three dimensions. In the opposite limit R to 0, crossover between d-1 and d-dimensional order occurs, and from the authors' analysis of the three-dimensional series it appears that here the crossover exponent phi _d-1 is not equal to the two-dimensional mean-size exponent.