Bond percolation for homogeneous two-dimensional lattices

Abstract

Abstract Bond percolation is studied for the three homogeneous two-dimensional lattices: square lattice (SL), triangular lattice (TL) and honeycomb lattice (HL). An expanding cell technique is used to obtain percolation thresholds and other relevant information for different cell sizes. We extend the analysis as to include slightly asymmetric cells in addition to the usual symmetric cells to get more points in the scaling analysis. Exact percolation functions are obtained for each size. Then, the percolation threshold is obtained by means of two complementary methods: one based on the well-known renormalization techniques and the other one introduced here which is based upon determining the inflection point of the percolation curves. A comparison of the results obtained by these two methods is performed. The study includes iterations to extrapolate numerical results towards the thermodynamic limit. Critical exponents ν , β and γ are obtained. Values are compared with numerical results and expected theoretical estimations; present results show agreement and even improvement (in the case of γ ) with respect to some numeric values available in the literature. Comparison tables are provided.