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Loop statistics for dimer models on the torus

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A dimer configuration of a graph G is a subset of edges such that every vertex of G is adjacent to exactly one edge of this subset. If we superimpose two dimer configurations of the same graph, we get double edges and loops. When G is a large piece of a periodic graph drawn on the torus, these loops can wind around the torus non trivially. We derive the limiting law, when the size of the mesh of G goes to zero, of the winding number of this family of loop when the two configurations are sampled at random from a Gibbs measure.

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