Communications in Mathematical Physics

, Volume 319, Issue 1, pp 69–110 | Cite as

From Cycle Rooted Spanning Forests to the Critical Ising Model: an Explicit Construction

  • Béatrice de TilièreEmail author


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 \({\mathcal{G}}\) of this graph (Fisher in J Math Phys 7:1776–1781, 1966). In this paper we explicitly relate the dimer model associated to the critical Ising model and critical cycle rooted spanning forests (CRSFs). This relation is established through characteristic polynomials, whose definition only depends on the respective fundamental domains, and which encode the combinatorics of the model. We first show a matrix-tree type theorem establishing that the dimer characteristic polynomial counts CRSFs of the decorated fundamental domain \({\mathcal{G}_1}\). Our main result consists in explicitly constructing CRSFs of \({\mathcal{G}_1}\) counted by the dimer characteristic polynomial, from CRSFs of G 1, where edges are assigned Kenyon’s critical weight function (Kenyon in Invent Math 150(2):409–439, 2002); thus proving a relation on the level of configurations between two well known 2-dimensional critical models.


Characteristic Polynomial Outgoing Edge Dual Graph Dime Model Oriented Edge 
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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Laboratoire de Probabilités et Modèles AléatoiresUniversité Pierre et Marie CurieParisFrance
  2. 2.Institut de MathématiquesUniversité de NeuchâtelNeuchâtelSwitzerland

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