Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder

• Published in: Journal of physics. A, Mathematical and theoretical Vol. 54; no. 22; pp. 22 - 35
• Main Author: Povolotsky, A M
• Format: Journal Article
• Language: English
• Published: IOP Publishing 04.06.2021
• Subjects: Baxter’s T–Q equation; loop models; percolation; six-vertex model
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8121/abf6fe

Abstract 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.