Renormalisation group treatment of bond percolation in anisotropic and 'inhomogeneous' planar lattices

The uncorrelated bond percolation problem is studied in three planar systems where there are two distinct occupancy probabilities. The authors apply two different real space renormalisation group approaches (referred as the 'canonical' (CRG) and the 'parametric' (PRG) ones) to the anisotropic first-neighbour square lattice, and both of them exhibit the expected tendency towards the exactly known phase boundary (p+q=1). Then they introduce, within the context of PRG calculations for increasingly large cells, an extrapolation method which leads to analytical proposals for the other two lattices, namely p+q= 1 / 2 for the first- and second-neighbour square lattice (p and q are respectively the first- and second-neighbour occupancy probabilities), and 3(p- 1 / 2 )=4((1-q) 2 +(1-q) 3 ) (p and q are respectively the occupancy probabilities of the topologically different bonds which are in a 1:2 ratio) for the 4-8 lattice.