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… 
1 Citations
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.

References

SHOWING 1-10 OF 67 REFERENCES
Stochastic Lowner Evolution and the Scaling Limit of Critical Models
Great progress in the understanding of conformally invariant scaling limits of stochastic models, has been given by the Stochastic Lowner Evolutions (SLE). This approach has been pioneered by Schramm
Coulomb Gas Formulation of Two Dimensional Kosterlitz-Thouless Phase Transition
The absence of conventional long range order in the classical two dimensional xy model is introduced. Two main types of excitations of this model, spin wave and vortices, are analyzed in the
Exactly solved models in statistical mechanics
exactly solved models in statistical mechanics exactly solved models in statistical mechanics rodney j baxter exactly solved models in statistical mechanics exactly solved models in statistical
Taiwan
Under what conditions would the democracies in Northeast Asia seek to join the nuclear weapons club? Japan, South Korea and Taiwan are threshold nuclear powers by virtue of their robust civilian
Conformal field theory applied to loop models Polygons, Polyominoes and Polyhedra (Lecture Notes in Physics vol 775) ed A
  • J Guttmann (Berlin: Springer)
  • 2009
Exact densities of loops in O (1) dense loop model and of clusters in critical percolation on a cylinder
  • Journal of Physics A: Mathematical and Theoretical,
  • 2021
The eight-vertex model and Painlevé VI equation II: eigenvector results
We study a special anisotropic -model on a periodic chain of an odd length and conjecture exact expressions for certain components of the ground state eigenvectors. The results are written in terms
Relations between the ‘percolation’ and ‘colouring’ problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ‘percolation’ problem
  • H. Temperley, E. Lieb
  • Mathematics
    Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
  • 1971
A transfer-matrix approach is introduced to calculate the 'Whitney polynomial’ of a planar lattice, which is a generalization of the ‘percolation’ and ‘colouring’ problems. This new approach turns
Residual Entropy of Square Ice
At low temperatures, ice has a residual entropy, presumably caused by an indeterminacy in the positions of the hydrogen atoms. While the oxygen atoms are in a regular lattice, each O-H-O bond permits
...
...