# 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}.
Document type :
Preprints, Working Papers, ...
51 pages, 24 figures. 2011
Domain :

https://hal.archives-ouvertes.fr/hal-00557878
Contributor : Béatrice De Tilière <>
Submitted on : Thursday, January 20, 2011 - 11:37:28 AM
Last modification on : Tuesday, May 30, 2017 - 1:07:46 AM

### Identifiers

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

### Citation

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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