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

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

2010-12-21

arxiv.org/abs/1012.4836v2

Physics

Mathematical Physics

51 pages, 24 figures

Scientific paper

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}.

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