论文标题
O（1）密集环模型中的环和晶体中的簇II中的簇II：旋转晶格的簇密度的确切密度
Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder II: rotated lattice
论文作者
论文摘要
这项工作继续进行研究，从\ cite {povolotsky2021}开始，其中获得了具有周期性边界条件的无限平方晶格条上O（1）密度的环路模型的确切密度。这些密度也等于四十五度旋转的方格上的临界渗滤簇的密度滚动到圆柱体中。在这里，我们将这些结果扩展到倾斜的方格。这特别使我们能够在以前进行了广泛研究的标准方向的平方晶格的圆柱体上获得临界渗滤簇的密度。我们获得了可缩度和不可缩的环的确切密度，或者等效地，临界渗透簇的密度分别在圆柱体周围并不包裹。该解决方案将O（1）密集环模型的映射到Razumov-Stroganov点中的六vertex模型，而有效倾斜是通过Fujimoto提出的不均匀转移矩阵引入的。进一步的解决方案是基于Bethe Ansatz和Baxter T-Q方程的弗里德金·史特罗诺夫·扎吉尔的解决方案。结果以两个线性代数方程的两个显式系统的解表示，可以分析圆柱体的小圆周或数值对较大圆柱的圆周进行。我们列出了密度在小圆周和几个晶格方向上的密度的确切合理值，并使用高精度数值计算的结果来研究有限尺寸的校正对密度的校正，特别是它们对晶格的倾斜度的依赖性。
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.
