  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.
