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 forty five 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.