Critical Ising Model and Cycle Rooted Spanning Forests: an explicit correspondence

Abstract : Fisher established an explicit correspondence between the 2-dimensional Ising model defined on a graph $G$ and the dimer model defined on a decorated version $\GD$ of this graph \cite{Fisher}. In this paper, we prove an explicit correspondence between the dimer model associated to a large class of critical Ising models whose underlying graph is periodic, and critical cycle rooted spanning forests (CRSFs). The correspondence is established through characteristic polynomials, whose definition only depends on the respective fundamental domains, and which encode the combinatorics of the models. We first show a matrix-tree type theorem establishing that the dimer characteristic polynomial counts CRSFs of the decorated fundamental domain $\GD_1$. Our main result consists in an explicit, weight preserving correspondence between CRSFs of $\GD_1$ counted by the dimer characteristic polynomial and CRSFs of $G_1$, where edges are assigned Kenyon's critical weight function \cite{Kenyon3}.
Type de document :
Pré-publication, Document de travail
51 pages, 24 figures. 2011
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https://hal.archives-ouvertes.fr/hal-00557878
Contributeur : Béatrice De Tilière <>
Soumis le : jeudi 20 janvier 2011 - 11:37:28
Dernière modification le : mercredi 12 octobre 2016 - 01:04:01

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  • HAL Id : hal-00557878, version 1
  • ARXIV : 1012.4836

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UPMC | INSMI | PMA | USPC

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Béatrice De Tilière. Critical Ising Model and Cycle Rooted Spanning Forests: an explicit correspondence. 51 pages, 24 figures. 2011. <hal-00557878>

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