BOND PERCOLATION PROBLEM IN A SEMI-INFINITE MEDIUM - LANDAU-GINZBURG THEORY

The bond percolation problem in a lattice bound by a plane surface is defined by associating different occupation probabilities PB, P, or P to bonds in the bulk, on the surface, or linking the bulk to the surface, respectively. The coordinate z indicates the distance of parallel planes to the surface. The z-dependent percolation probability (z) is defined as the probability that a site on the plane z belongs to an infinite cluster and I derive a rigorous expression for (z) by introducing z-dependent external fields hz in the q states Potts Hamiltonian in the limit q=1. Gaussian integration techniques are used to derive a Landau-Ginzburg free-energy functional for arbitrary q. The resulting differential equations for the order parameter are explicitly solved for the bond percolation problem. We introduce the parameters t=2nln[qBqc] and =2ln[qnsqqcn], where q=1-P is the probability of a bond being absent, and n(ns) is the coordination number in the bulk (surface). Also Pc=1-exp(-1n) is the mean-field value of the percolation concentration of bonds in the bulk. We obtain the following results: (a) for t>0, s<<t, where s=-12(tn)12, all clusters are finite; (b) for t>0, <s, all clusters are finite in the bulk but the probability for an infinite cluster to form at and near the surface is nonvanishing; (c) for <t<0 an infinite cluster forms through the whole system, but it has a larger probability of being close to the surface; (d) for 0<t< only finite clusters occur in the system; (e) for >t and t<0, an infinite cluster starts to form in the whole system, with a larger probability of being in the bulk than on the surface. Critical exponents are derived for the different transitions and they are shown to satisfy general scaling relations. © 1979 The American Physical Society.