Analytic and Computational Analysis for Bond Percolation on Two‐Dimensional Lattices

Beginning in the 60t’s it has been a growing number of studies referred to disordered systems. Percolation theory has been one of the most popular techniques applied to solving these problems. We report here an analytical method to deal with percolation in two‐dimensional lattices. In particular, we inform results on bond percolation related to square, triangular and honeycomb lattices, selecting different types of cells associated to each geometry. Thus, we choose two cells related to square lattices whose sizes are 5 [1] and 13 [2] and two cells associated to triangular lattices with sizes 7 and 19 respectively. For the case of honeycomb lattices we have considered one cell of size 15. For each cell, we calculate probabilities of generating bond percolation along one direction. We must include all possible configurations associated to each cell, where p represents an occupied bond and 1 − p represents an empty bond, according to usual treatments [1]. Percolation function, p′, is expressed as a polynomial in terms of p, whose critical value permits to calculate the percolation threshold p_c, and critical parameter v, according to usual definitions. On the other hand, these analytical functions are compared with results of numerical simulations reported here as well as numerical data available in the literature [3,4].