## Conformal invariance of loops in the double-dimer model.(English)Zbl 1283.05218

Summary: The dimer model is the study of random dimer covers (perfect matchings) of a graph. A double-dimer configuration on a graph $$\mathcal G$$ is a union of two dimer covers of $$\mathcal G$$. We introduce quaternion weights in the dimer model and show how they can be used to study the homotopy classes (relative to a fixed set of faces) of loops in the double dimer model on a planar graph. As an application we prove that, in the scaling limit of the “uniform” double-dimer model on $$\mathbb Z^2$$ (or on any other bipartite planar graph conformally approximating $$\mathbb C$$), the loops are conformally invariant.
As other applications we compute the exact distribution of the number of topologically nontrivial loops in the double-dimer model on a cylinder and the expected number of loops surrounding two faces of a planar graph.

### MSC:

 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C10 Planar graphs; geometric and topological aspects of graph theory

### Keywords:

random dimer covers; perfect matchings
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### References:

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