Honeycomb lattices with defects

In this paper, we introduce a variant of the honeycomb lattice in which we create defects by randomly exchanging adjacent bonds, producing a random tiling with a distribution of polygon edges. We study the percolation properties on these lattices as a function of the number of exchanged bonds using an alternative computational method. We find the site and bond percolation thresholds are consistent with other three-coordinated lattices with the same standard deviation in the degree distribution of the dual; here we can produce a continuum of lattices with a range of standard deviations in the distribution. These lattices should be useful for modeling other properties of random systems as well as percolation.

Figure 1

Two examples of defective honeycomb (DHC) lattices. Each lattice has 16 tiles and periodic boundary conditions. Red tiles are rectangles, yellow tiles are pentagons, blue tiles are hexagons, green tiles are heptagons, and purple tiles are octagons. (a) Example of a DHC lattice with standard deviation in degree of the dual $\sigma =0.5$. (b) Example of a DHC lattice with $\sigma =1.658$.

Figure 2

Pictorial representation of the flip of bond $a$–$b$ in the DHC lattice. The bonds between vertices 1, 2, 3, 4, $a$, and $b$ are rearranged such that tiles $C$ and $D$ become neighbors instead of $A$ and $B$. (a) Correct flip. The bond 2–$a$ becomes 2–$b$ and the bond 4–$b$ becomes 4–$a$. The lattice remains planar and the bond $a$–$b$ separates two new tiles. (b) Incorrect flip. There is no way to arrange sites $a$ and $b$ in the plane such that none of the bonds cross.

Figure 3

Pictures of the generation of an $n=4, f=10/48=0.21$ lattice. Ten distinct edges are flipped one at a time to generate the final lattice. Each panel shows the state of the lattice after each edge flip. The flipped edge is highlighted in red. The sites and tiles are labeled in blue and red, respectively, as described in Appendix pp1.

Figure 4

Relationship between the variance in the number of edges per tile and the fraction of bonds flipped $f$. (a) Orange disks represent the variance found in 200 independently generated $128\times 128$ tile lattices. Red triangles represent the variance found in 50 independently generated $512\times 512$ tile lattices. Black circles represent the eight $1500\times 1500$ tile lattices used to find the percolation thresholds in Sec. 5. The data can be fit by the function $v=-4.835f^2+11.92f+0.0681$. (b) Average distribution of polygons over ten independent $128\times 128$ lattices at different flipping fractions. As more bonds are flipped, a greater percentage of the tiles are triangles and many-sided polygons.

Figure 5

Graph of the relationship between the percolation threshold and dual-lattice variance $v={\sigma }^2$. The bond (lower) and site (upper) thresholds for the DHC lattice and the previously studied honeycomb variant lattices listed in Table 2 are shown. The linear regression for the bond thresholds is $p_c=0.00649v+0.6533$ and for the site threshold it is $p_c=0.00783v+0.6978$.

Figure 6

Example of finding the percolation threshold using our method. We grew 10 000 clusters on a honeycomb lattice at various occupation probabilities $p$. Statistics on the average $P_{\ge s}$ and $\langle s^{\prime }\le s\rangle$ were gathered and the ratio $y=s{P}_{\ge s}/\langle s^{\prime }\le s\rangle$ was plotted vs the cluster size $s$. At the percolation threshold, the resulting curve should be flat. In (a) we get the approximate value of $p_c$ for site percolation on the honeycomb lattice. In (b) we narrow in on the exact value. Using this method, we determined that the site percolation threshold was $0.69700\pm 0.00005$, which agrees with the known value of 0.697 040 [3].