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
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.
