Percolation in two-dimensional lattices. I. A technique for the estimation of thresholds.

Abstract

We put forward a relatively simple method to estimate reliable percolation thresholds ${p}_{c}$ of two-dimensional (2D) lattices both for the bond and site problem. On the basis of this method, we actually evaluate ${p}_{c}$ for several 2D lattices by analyzing the results which we obtain from Monte Carlo simulations. Our method enables us to achieve three significant figures for ${p}_{c}$ even when the system size N is less than 300\ifmmode\times\else\texttimes\fi{}300 and the increment \ensuremath{\Delta}p of the concentration p of intact bonds or sites is 0.002. We ascertain that our method works both for the bond and site problem, both for periodic and nonperiodic lattices, and both for lattices with single-valued and mixed-valued coordination.