Precise determination of the bond percolation thresholds and finite-size scaling corrections for the sc, fcc, and bcc lattices

Abstract

Extensive Monte Carlo simulations were performed to study bond percolation on the simple cubic (sc), face-centered-cubic (fcc), and body-centered-cubic (bcc) lattices, using an epidemic approach. These simulations provide very precise values of the critical thresholds for each of the lattices: ${p}_{c}(\mathrm{sc})=0.2488126\ifmmode\pm\else\textpm\fi{}0.0000005,$ ${p}_{c}(\mathrm{fcc})=0.1201635\ifmmode\pm\else\textpm\fi{}0.0000010,$ and ${p}_{c}(\mathrm{bcc})=0.1802875\ifmmode\pm\else\textpm\fi{}0.0000010$. For $p$ close to ${p}_{c},$ the results follow the expected finite-size and scaling behavior, with values for the Fisher exponent \ensuremath{\tau} $(2.189\ifmmode\pm\else\textpm\fi{}0.002),$ the finite-size correction exponent \ensuremath{\Omega} $(0.64\ifmmode\pm\else\textpm\fi{}0.02),$ and the scaling function exponent \ensuremath{\sigma} $(0.445\ifmmode\pm\else\textpm\fi{}0.01)$ confirmed to be universal.