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

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.