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

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 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 $\GD_1$. Our main result consists in explicitly constructing CRSFs of $\GD_1$ counted by the dimer characteristic polynomial, from CRSFs of $G_1$ where edges are assigned Kenyon's critical weight function \cite{Kenyon3}; thus proving a relation on the level of configurations between two well known 2-dimensional critical models.