• DOI:10.1103/PhysRevE.92.032134
• Corpus ID: 34100812

Dimer covering and percolation frustration.

@article{HajiAkbari2015DimerCA,
  title={Dimer covering and percolation frustration.},
  author={Amir Haji-Akbari and Nasim Haji-Akbari and Robert M. Ziff},
  journal={Physical review. E, Statistical, nonlinear, and soft matter physics},
  year={2015},
  volume={92 3},
  pages={
          032134
        }
}

• A. Haji-Akbari, Nasim Haji-Akbari, R. Ziff
• Published 15 July 2015
• Computer Science, Mathematics
• Physical review. E, Statistical, nonlinear, and soft matter physics

Covering a graph or a lattice with nonoverlapping dimers is a problem that has received considerable interest in areas, such as discrete mathematics, statistical physics, chemistry, and materials science. Yet, the problem of percolation on dimer-covered lattices has received little attention. In particular, percolation on lattices that are fully covered by nonoverlapping dimers has not evidently been considered. Here, we propose a procedure for generating random dimer coverings of a given lattice. We then compute the bond percolation threshold on random and ordered coverings of the square and the triangular lattices on the remaining bonds connecting the dimers. We obtain p_{c}=0.367713(2) and p_{c}=0.235340(1) for random coverings of the square and the triangular lattices, respectively. We observe that the percolation frustration induced as a result of dimer covering is larger in the low-coordination-number square lattice. There is also no relationship between the existence of long-range order in a covering of the square lattice and its percolation threshold. In particular, an ordered covering of the square lattice, denoted by shifted covering in this paper, has an unusually low percolation threshold and is topologically identical to the triangular lattice. This is in contrast to the other ordered dimer coverings considered in this paper, which have higher percolation thresholds than the random covering. In the case of the triangular lattice, the percolation thresholds of the ordered and random coverings are very close, suggesting the lack of sensitivity of the percolation threshold to microscopic details of the covering in highly coordinated networks.