Haji-Akbari Amir, Haji-Akbari Nasim, Ziff Robert M
Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08544, USA.
Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Sep;92(3):032134. doi: 10.1103/PhysRevE.92.032134. Epub 2015 Sep 24.
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.
用不重叠的二聚体覆盖图或晶格是一个在离散数学、统计物理、化学和材料科学等领域备受关注的问题。然而，关于二聚体覆盖晶格上的渗流问题却很少受到关注。特别是，对于完全由不重叠二聚体覆盖的晶格上的渗流，显然尚未得到考虑。在此，我们提出一种生成给定晶格随机二聚体覆盖的方法。然后，我们计算正方形和三角形晶格在连接二聚体的剩余键上的随机和有序覆盖的键渗流阈值。对于正方形和三角形晶格的随机覆盖，我们分别得到(p_{c}=0.367713(2))和(p_{c}=0.235340(1))。我们观察到，由于二聚体覆盖导致的渗流挫折在低配位数的正方形晶格中更大。正方形晶格覆盖中的长程序存在与否与其渗流阈值之间也没有关系。特别是，本文中表示为移位覆盖的正方形晶格的有序覆盖具有异常低的渗流阈值，并且在拓扑上与三角形晶格相同。这与本文中考虑的其他有序二聚体覆盖形成对比，后者的渗流阈值高于随机覆盖。在三角形晶格的情况下，有序和随机覆盖的渗流阈值非常接近，这表明在高度配位的网络中，渗流阈值对覆盖的微观细节缺乏敏感性。
