Entropy of fully packed hard rigid rods on $d$-dimensional hyper-cubic lattices

TL;DR

This paper analyzes the asymptotic entropy of fully packed rigid rods on large square lattices, revealing a universal large-$k$ behavior and exploring configuration connectivity via flip moves.

Contribution

It introduces a detailed analysis of the entropy scaling for large rods and conjectures a super-universal behavior across all higher-dimensional hyper-cubic lattices.

Findings

Entropy per site scales as $A k^{-2} \\ln k$ with $A=1$ for large $k$

Full coverage configurations are connected via flip moves from a standard parallel configuration

The large-$k$ entropy behavior is conjectured to be super-universal in all $d$-dimensional hyper-cubic lattices

Abstract

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 $S_2(k)$ tends to $ A k^{-2} \ln k$, with $A=1$. We conjecture, based on a perturbative series expansion, that this large-$k$ behavior of entropy per site is super-universal and continues to hold on all $d$-dimensional hyper-cubic lattices, with $d \geq 2$.