Random site percolation on honeycomb lattices with complex neighborhoods

Abstract

We present a rough estimation -- up to four significant digits, based on the scaling hypothesis and the probability of belonging to the largest cluster vs. the occupation probability -- of the critical occupation probabilities for the random site percolation problem on a honeycomb lattice with complex neighborhoods containing sites up to the fifth coordination zone. There are 31 such neighborhoods with their radius ranging from one to three and containing from three to 24 sites. For two-dimensional regular lattices with compact extended-range neighborhoods, in the limit of the large number $z$ of sites in the neighborhoods, the site percolation thresholds $p_c$ follow the dependency $p_c\propto 1/z$, as recently shown by Xun, Hao and Ziff [Physical Review E 105, 024105 (2022)]. On the contrary, noncompact neighborhoods (with holes) destroy this dependence due to the degeneracy of the percolation threshold (several values of $p_c$ corresponding to the same number $z$ of sites in the neighborhoods). An example of a single-value index $\zeta ={\sum }_i{z}_i{r}_i$ -- where $z_i$ and $r_i$ are the number of sites and radius of the $i$-th coordination zone, respectively -- characterizing the neighborhood and allowing avoiding the above-mentioned degeneracy is presented. The percolation threshold obtained follows the inverse square root dependence $p_c\propto 1/\sqrt{\zeta }$. The functions boundaries() (written in C) for basic neighborhoods (for the unique coordination zone) for the Newman and Ziff algorithm [Physical Review E 64, 016706 (2001)] are also presented.

11 pages, 4 figures, 1 table & three appendices with 2 extra figures and 5 procedures written in C

References in corpus (15)

• Recent advances in percolation theory and its applications
• Percolation on complex networks: Theory and application
• Square lattice site percolation at increasing ranges of neighbor interactions
• Exact bond percolation thresholds in two dimensions
• Site percolation on square and simple cubic lattices with extended neighborhoods and their continuum limit
• Simple cubic random-site percolation thresholds for neighborhoods containing fourth-nearest neighbors
• Percolation of overlapping squares or cubes on a lattice
• Universal behavior of site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in 2 and 3 dimensions
• Critical polynomials in the nonplanar and continuum percolation models
• Critical frontier for the Potts and percolation models on triangular-type and kagome-type lattices II: Numerical analysis
• Percolation in Networks with Voids and Bottlenecks
• Restoring site percolation on a damaged square lattice
• Percolation thresholds on triangular lattice for neighbourhoods containing sites up-to the fifth coordination zone
• Square lattice site percolation thresholds for complex neighbourhoods
• Exact percolation probabilities for a square lattice: Site percolation on a plane, cylinder, and torus