We obtain exact densities of contractible and non-contractible loops in the O(1) model on a strip of the square lattice rolled into an infinite cylinder of finite even circumference L. They are also equal to the densities of critical percolation clusters on 45 degree rotated square lattice rolled into a cylinder, which do not or do wrap around the cylinder respectively. The results are presented as explicit rational functions of L taking rational values for any even L. Their asymptotic expansions in the large L limit have irrational coefficients reproducing the earlier results in the leading orders. The solution is based on a mapping to the six-vertex model and the use of technique of Baxter’s T–Q equation.

我们在O (1) 模型中获得了可收缩和不可收缩环的精确密度，该模型将方格条卷成有限偶数周长L的无限圆柱体。它们也等于卷成圆柱体的 45 度旋转方形晶格上的临界渗流簇的密度，它们分别不缠绕或确实缠绕在圆柱体周围。结果表示为 L 的显式有理函数，对任何偶数 L取有理值。它们在大L极限中的渐近展开具有无理系数，再现了前导顺序中的早期结果。该解决方案基于到六顶点模型的映射和使用 Baxter 的 T-Q 方程技术。