Geodesic
Aperiodic non-isomorphic lattices with equivalent percolation and random-cluster models
The electronic journal of combinatorics, Tome 17 (2010)
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We explicitly construct an uncountable class of infinite aperiodic plane graphs which have equal, and explicitly computable, bond percolation thresholds. Furthermore for both bond percolation and the random-cluster model all large scale properties, such as the values of the percolation threshold and the critical exponents, of the graphs are equal. This equivalence holds for all values of $p$ and all $q\in[0,\infty]$ for the random-cluster model. The graphs are constructed by placing a copy of a rotor gadget graph or its reflection in each hyperedge of a connected self-dual 3-uniform plane hypergraph lattice. The exact bond percolation threshold may be explicitly determined as the root of a polynomial by using a generalised star-triangle transformation. Related randomly oriented models share the same bond percolation threshold value.
DOI : 10.37236/320
Classification : 60K35, 05C80, 05C65, 05C63
Mots-clés : Bond percolation threshold, plane lattice, plane hypergraph lattice, random-cluster model 
@article{10_37236_320,
     author = {Klas Markstr\"om and John C. Wierman},
     title = {Aperiodic non-isomorphic lattices with equivalent percolation and random-cluster models},
     journal = {The electronic journal of combinatorics},
     year = {2010},
     volume = {17},
     doi = {10.37236/320},
     zbl = {1200.60086},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/320/}
}
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%A Klas Markström
%A John C. Wierman
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Klas Markström; John C. Wierman. Aperiodic non-isomorphic lattices with equivalent percolation and random-cluster models. The electronic journal of combinatorics, Tome 17 (2010). doi: 10.37236/320

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