O(1)稠密环模型中的环密度与圆柱上临界逾渗团密度的精确结果 II：旋转晶格
Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder II: rotated lattice
AI总结：
本研究将O(1)稠密环模型在倾斜方形晶格上的环密度精确解推广到标准取向晶格圆柱，给出临界逾渗团簇的精确密度，并分析有限尺寸修正。
摘要：
本工作继续了在\cite{Povolotsky2021}中开始的研究，其中获得了方形晶格无限条带上具有周期性边界条件的O(1)稠密环模型中环的精确密度。这些密度也等于四十五度旋转方形晶格卷成圆柱时临界逾渗团簇的密度。在此，我们将这些结果扩展到具有倾斜的方形晶格。这尤其使我们能够获得之前广泛研究的标准取向方形晶格圆柱上临界逾渗团簇的密度。我们获得了可收缩和不可收缩环的精确密度，或者等价地，分别不环绕和环绕圆柱的临界逾渗团簇的密度。解决方案利用了O(1)稠密环模型到Razumov-Stroganov点处六顶点模型的映射，而有效倾斜通过Fujimoto提出的非均匀转移矩阵引入。进一步的解决方案基于Bethe ansatz和Fridkin-Stroganov-Zagier对Baxter的T-Q方程的解。结果以两个显式线性代数方程组的解表示，对于小周长的圆柱可以解析求解，对于较大周长可以数值求解。我们给出了小周长圆柱和几种晶格取向的精确有理密度值，并使用高精度数值计算结果研究密度的有限尺寸修正，特别是它们对晶格倾斜的依赖性。
英文摘要：
This work continues the study started in \cite{Povolotsky2021}, where the exact densities of loops in the O(1) dense loop model on an infinite strip of the square lattice with periodic boundary conditions were obtained. These densities are also equal to the densities of critical percolation clusters on the forty five degree rotated square lattice rolled into a cylinder. Here, we extend those results to the square lattice with a tilt. This in particular allow us to obtain the densities of critical percolation clusters on the cylinder of the square lattice of standard orientation extensively studied before. We obtain exact densities of contractible and non-contractible loops or equivalently the densities of critical percolation clusters, which do not and do wrap around the cylinder respectively. The solution uses the mapping of O(1) dense loop model to the six-vertex model in the Razumov-Stroganov point, while the effective tilt is introduced via the the inhomogeneous transfer matrix proposed by Fujimoto. The further solution is based on the Bethe ansatz and Fridkin-Stroganov-Zagier's solution of the Baxter's T-Q equation. The results are represented in terms of the solution of two explicit systems of linear algebraic equations, which can be performed either analytically for small circumferences of the cylinder or numerically for larger ones. We present exact rational values of the densities on the cylinders of small circumferences and several lattice orientations and use the results of high precision numerical calculations to study the finite-size corrections to the densities, in particular their dependence on the tilt of the lattice.
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