Exact loop densities in $O(1)$ dense loop model on the cylinder of odd circumference and clusters in half-turn self-dual critical percolation

TL;DR

This paper calculates exact loop densities in an $O(1)$ dense loop model on a cylindrical lattice with odd circumference, revealing a connection to critical percolation clusters in a rotated lattice with special boundary conditions.

Contribution

It provides the first exact calculation of loop densities in this specific $O(1)$ dense loop model on a cylinder of odd circumference, linking it to half-turn self-dual percolation.

Findings

Exact loop densities are derived for the model.

Loop densities match cluster densities in a rotated percolation lattice.

The solution uses a correspondence with the six-vertex model at a special point.

Abstract

We consider $O(1)$ dense loop model in a square lattice wrapped on a cylinder of odd circumference $L$ and calculate the exact densities of loops. These densities of loops are equal to the densities of critical bond percolation clusters on a forty-five-degree rotated square lattice rolled into a cylinder with special boundary conditions which we refer to as half-turn self-dual percolation. The solution is based on a correspondence between the $O(1)$ dense loop model and the six-vertex model at the Razumov-Stroganov point.