Dimer Covering and Percolation Frustration

TL;DR

This paper investigates how covering lattices with dimers affects percolation thresholds, revealing that dimer covering induces frustration and that ordered coverings can significantly alter percolation properties.

Contribution

It introduces a new method for generating random dimer coverings and analyzes their impact on percolation thresholds on square and triangular lattices, highlighting the effects of order and lattice coordination.

Findings

Percolation thresholds are approximately 0.368 for square and 0.235 for triangular lattices.

Dimer covering induces greater percolation frustration in low-coordination lattices.

Ordered coverings can have lower or similar thresholds compared to random coverings, depending on lattice topology.

Abstract

Covering a graph or a lattice with non-overlapping 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 non-overlapping dimers has not evidently been considered. Here, we propose a novel 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 lattice, 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 lattice, 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 work, 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 work, 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.