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

@article{Povolotsky2021ExactDO,
  title={Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder},
  author={A. M. Povolotsky},
  journal={Journal of Physics A: Mathematical and Theoretical},
  year={2021},
  volume={54}
}
  • A. M. Povolotsky
  • Published 28 January 2021
  • Mathematics
  • Journal of Physics A: Mathematical and Theoretical
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… 
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Exact results for average cluster numbers in bond percolation on infinite-length lattice strips.
TLDR
It is proved that exact analytic expressions for the average cluster numbers 〈k〉_{Λ_{s}} on infinite-length strips Κ, with various widths, of several different lattices, are rational functions of p, and that they determine the radii of convergence of series expansions for small p and for p near to unity.

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