AI 中文总结
研究\(\mathbb{Z}^2\)上硬矩形系统中向列相形成所需纵横比，采用双尺度团簇展开方法，明确相关常数和参数，完善论证，得出\(k\geq10^{72}\)这一严格估计，是该问题的首个此类结果。
AI 中文摘要
我们证明了在具有两种允许取向且纵横比\(k := l/w\)较大的正方形晶格上\(l\times w\)硬矩形模型中向列相的存在。证明基于Disertori - Giuliani先前为二维硬棒以及Disertori - Giuliani - Jauslin为三维硬板开发的双尺度团簇展开方法。我们的主要贡献在于明确追踪常数和参数，并完善这些工作中隐含的论证。由此证明产生了一组充分的定量条件，从中可提取所需纵横比的估计值。对这些条件的非优化评估得出界限\(k\geq10^{72}\)。尽管它远超过数值预测\(k_{\min}=7\)，但我们的结果似乎是硬矩形系统中向列相形成所需纵横比的首个严格估计。
英文摘要
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 $k:=l/w$. 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.
Comments45 pages
