Anisotropic bond percolation

We introduce anisotropic bond percolation in which there exist different occupation probabilities for bonds placed in different coordinate directions. We study in detail a d-dimensional hypercubical lattice, with probabilities p I for bonds within (d 1)-dimensional layers perpendicular to the z direction, and p11= Rp, for bonds parallel to z . For this model, we calculate low-density series for the mean size S, in both two and three dimensions for arbitrary values of the anisotropy parameter R. We find that in the limit 1/R + 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, we derive that the crossover exponent (=&) 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 + 0, crossover between d 1 and d-dimensional order occurs, and from our analysis of the three-dimensional series it appears that here the crossover exponent & I is not equal to the two-dimensional mean-size exponent. This feature is in contrast with the corresponding situation in thermal critical phenomena where 6 d l does equal the susceptibility exponent in two dimensions. Finally, our analysis appears to confirm that the value of the mean-size exponent is independent of anisotropy in accordance with universality.