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arXiv · 2101.12096

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

A. M. Povolotsky
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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.

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A. M. Povolotsky. 2021-01-28. Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder. https://doi.org/10.1088/1751-8121%2Fabf6fe

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