Convergence of Mean-Field Approximations in Site Percolation and Application of CAM to d— 1 Further-Neighbors Percolation Problem

We study the mean-field approximation in the site-percolation problem. Using the analog of the Simon-Lieb inequality, we show that the mean-field critical probability is convergent to the exact value when the size of clusters tends to infinity. Applying this approximation to the one-dimensional further-neighbor percolation problem and calculating some critical coefficients, we prove that the asymptotic scaling relations predicted by the coherent-anomaly method are satisfied.