Series expansions of the percolation probability on the directed triangular lattice

We have derived long-series expansions of the percolation probability for site, bond and site-bond percolation on the directed triangular lattice. For the bond problem we have extended the series from order 12 to 51 and for the site problem from order 12 to 35. For the site-bond problem, which has not been studied before, we have derived the series to order 32. Our estimates of the critical exponent are in full agreement with results for similar problems on the square lattice, confirming expectations of universality. For the critical probability and exponent we find in the site case: qc D 0:404 352 8 0:000 001 0 and D 0:276 45 0:000 10; in the bond case: qc D 0:521 98 0:000 01 and D 0:2769 0:0010; and in the site-bond case: qc D 0:264 173 0:000 003 and D 0:2766 0:0003. In addition we have obtained accurate estimates for the critical amplitudes. In all cases we find that the leading correction to scaling term is analytic, i.e. the confluent exponent 1 D 1. In an earlier paper (Jensen and Guttmann 1995) we reported on the derivation and analysis of long series for the percolation probability of site and bond percolation on the directed square and hexagonal lattices. In this paper we extend this work to site, bond and site- bond percolation on the directed triangular lattice. We refer to our earlier paper for a more general introduction to directed percolation and its role in the modelling of physical systems. In directed site percolation each site is either present (with probability p) or absent (with probability q D 1 p) independent of all other sites on the lattice. Similarly for bond percolation each bond is absent or present independently of other bonds. Finally in site- bond percolation both sites and bonds may be absent or present with equal probability, but again with no dependency on any other sites or bonds. Two sites in the various models are connected if one can find a path, respecting the directions indicated in figure 1, through occupied sites, bonds or sites and bonds, respectively, from one to the other. When p is smaller than a critical value pc all clusters of connected sites remain finite, while for p > pc there is an infinite cluster spanning the lattice in the preferred direction. The order parameter of the system is the percolation probability P. p/that a given site belongs to the infinite cluster. This quantity is strictly zero when p p c the behaviour of P. p/in the vicinity of pc may be described by a critical exponent , P. p/ /.p pc/ p ! p C c : (1)