Percolation critical polynomial as a graph invariant.

@article{Scullard2011PercolationCP,
  title={Percolation critical polynomial as a graph invariant.},
  author={Christian R Scullard},
  journal={Physical review. E, Statistical, nonlinear, and soft matter physics},
  year={2011},
  volume={86 4 Pt 1},
  pages={
          041131
        }
}
  • C. Scullard
  • Published 4 November 2011
  • Mathematics
  • Physical review. E, Statistical, nonlinear, and soft matter physics
Every lattice for which the bond percolation critical probability can be found exactly possesses a critical polynomial, with the root in [0,1] providing the threshold. Recent work has demonstrated that this polynomial may be generalized through a definition that can be applied on any periodic lattice. The polynomial depends on the lattice and on its decomposition into identical finite subgraphs, but once these are specified, the polynomial is essentially unique. On lattices for which the exact… 

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References

C: Solid State Phys

  • 15, L757
  • 1982