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 $P_B$, $P_{\parallel }$, or $P_{\perp }$ 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 $\varphi \mathrm{}\left(z\right)\mathrm{}$ is defined as the probability that a site on the plane $z$ belongs to an infinite cluster and I derive a rigorous expression for $\varphi \mathrm{}\left(z\right)\mathrm{}$ by introducing $z$-dependent external fields $h_z$ in the $q$ states Potts Hamiltonian in the limit $q=1\mathrm{}$. 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 $t=2n\ln \left[\frac{q_B}{q_c}\right]$ and $\omega =2\mathrm{}\ln \left[\frac{q_{\parallel }^{\mathrm{ns}}{q}_{\perp }}{q_c^n}\right]$, where $q_{\alpha }=1-P_{\alpha }$ is the probability of a bond being absent, and $n\left(n_s\right)$ is the coordination number in the bulk (surface). Also $P_c=1-\exp (-\frac{1}{n})$ is the mean-field value of the percolation concentration of bonds in the bulk. We obtain the following results: (a) for $t>\mathrm{}0\mathrm{}$, ${\omega }_s<\omega <t$, where ${\omega }_s=-\frac{1}{2}{\left(\frac{t}{n}\right)}^{\frac{1}{2}}$, all clusters are finite; (b) for $t>\mathrm{}0\mathrm{}$, $\omega <{\omega }_s$, 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 $\omega <t<\mathrm{}0\mathrm{}$ an infinite cluster forms through the whole system, but it has a larger probability of being close to the surface; (d) for $0<t<\mathrm{}\omega$ only finite clusters occur in the system; (e) for $\omega >t$ and $t<\mathrm{}0\mathrm{}$, 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.