We determine the asymptotic behavior of the entropy of full coverings of a $L\times M$ square lattice by rods of size $k\times 1$ and $1\times k$, in the limit of large $k$. We show that full coverage is possible only if at least one of $L$ and $M$ is a multiple of $k$, and that all allowed configurations can be reached from a standard configuration of all rods being parallel, using only basic flip moves that replace a $k\times k$ square of parallel horizontal rods by vertical rods, and vice versa. In the limit of large $k$, we show that the entropy per site $S_2\left(k\right)$ tends to $A{k}^{-2}lnk$, with $A=1$. We conjecture, based on a perturbative series expansion, that this large-$k$ behavior of entropy per site is superuniversal and continues to hold on all $d$-dimensional hypercubic lattices, with $d\ge 2$.

• Received 13 December 2020
• Revised 19 February 2021
• Accepted 22 March 2021

DOI:https://doi.org/10.1103/PhysRevE.103.042130