Using Monte Carlo methods and finite-size scaling, we investigate surface critical phenomena in the bond-percolation model on the simple-cubic lattice with two open surfaces in one direction. We decompose the whole lattice into percolation clusters and sample the surface and bulk dimensionless ratios $Q_1$ and $Q_b$, defined on the basis of the moments of the cluster-size distribution. These ratios are used to determine critical points. At the bulk percolation threshold $p_{\mathit{bc}}$, we determine the surface bond-occupation probability at the special transition as $p_{1c}^{\left(\mathrm{s}\right)}=0.41817\left(2\right)$, and further obtain the corresponding surface thermal and magnetic exponents as $y_{t1}^{\left(\mathrm{s}\right)}=0.5387\left(2\right)$ and $y_{h1}^{\left(\mathrm{s}\right)}=1.8014\left(6\right)$, respectively. Next, from the pair correlation function on the surfaces, we determine $y_{h1}^{\left(\mathrm{o}\right)}=1.0246\left(4\right)$ and $y_{h1}^{\left(\mathrm{e}\right)}=1.25\left(6\right)$ for the ordinary and the extraordinary transition, respectively, of which the former is consistent with the existing result $y_{h1}^{\left(\mathrm{o}\right)}=1.024\left(4\right)$. We also numerically derive the line of surface phase transitions occurring at $p_b<p_{\mathit{bc}}$, and determine the pertinent asymptotic values of the universal ratios $Q_1$ and $Q_b$.

• Received 19 October 2004

DOI:https://doi.org/10.1103/PhysRevE.71.016117