Probability renormalisation group treatment of bond percolation in square, cubic and hypercubic lattices

Abstract

By using the real space renormalisation group method proposed by Reynolds, Klein and Stanley (see J. Phys. A, vol.11, p.L199 (1978)) the authors treat bond percolation on d-dimensional cubic lattices and obtain (through various extrapolation methods): (i) pc=1/2 (exact), nu p=1.351+0.012(-0.020) and alpha p=-0.700+0.040(-0.024) for the first-neighbour square lattice; (ii) pc=0.250+or-0.003 for the first and second-neighbour square lattice; (iii) pc=0.2526+or-0.0013, nu p=0.840+or-0.020 and alpha p=-0.520+or-0.060 for the first-neighbour cubic lattice; (iv) pc=0.149+or-0.010, nu p=0.667+or-0.030 and alpha p=-0.67+or-0.12 for the first-neighbour four-dimensional hypercubic lattice. Whenever comparison is possible these figures agree fairly well with other available results. The 'magnetic' scaling power yh is discussed for the square lattice. The influence on pc and nu p of the symmetry of the cluster and of the 'direction' of percolation is exhibited through several two-dimensional examples.