Bond percolation in distorted square and triangular lattices

This article presents a Monte Carlo study on bond percolation in distorted square and triangular lattices. The distorted lattices are generated by dislocating the sites from their regular positions. The amount and direction of the dislocations are random, but can be tuned by the distortion parameter α. Once the sites are dislocated, the bond lengths δ between the nearest neighbors change. A bond can only be occupied if its bond length is less than a threshold value called the connection threshold d. It is observed that when the connection threshold is greater than the lattice constant (assumed to be 1), the bond percolation threshold p_b always increases with distortion. For dle 1, no spanning configuration is found for the square lattice when the lattice is distorted, even very slightly. On the other hand, the triangular lattice not only spans for dle 1, it also shows a decreasing trend for p_b in the low-α range. These variation patterns have been linked with the average coordination numbers of the distorted lattices. A critical value d_c for the connection threshold has been defined as the value of d below which no spanning configuration can be found even after occupying all the bonds satisfying the connection criterion δle d. The behavior of d_c(α) is markedly different for the two lattices.