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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
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