Rigorous bound on the aspect ratio for the formation of a nematic phase in hard rod and hard rectangle systems on ${\mathbb{Z}}^2$

We prove the existence of a nematic phase in a model of $l\times w$ hard rectangles on the square lattice with two allowed orientations and a large aspect ratio $k:=l\mathrm{/}w$. The proof is based on a two-scale cluster expansion method developed previously by Disertori–Giuliani for hard rods in 2D and Disertori–Giuliani–Jauslin for hard plates in 3D. Our main contributions lie in explicitly tracking the constants and parameters and completing the arguments left implicit in these works. Hence, the proof produces a sufficient set of quantitative conditions from which estimates for the required aspect ratio can be extracted. A non-optimized evaluation of these conditions yields the bound $k\ge {10}^{72}$. Although it vastly overshoots the numerical prediction, $k_{\min }=7$, our result appears to be the first rigorous estimate of the aspect ratio required for the formation of a nematic phase in hard rectangle systems.