Renormalization Group Treatment of Bond Percolation in Anisotropic and 'Inhomogeneous' Planar Lattices

摘要

The uncorrelated bond percolation problem is studied in three planar systems where there are two distinct occupancy probabilities. Two different real space renormalization group approaches (referred as the 'canonical' (CRG) and the 'parametric' (PRG) ones) are applied 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, within the context of PRG calculations for increasingly large cells, an extrapolation method is introduced, which leads to analytic 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) exp 2 + (1-q) exp 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. (Atomindex citation 13:643007)