the double random current nesting field.

Summary: We relate the planar random current representation introduced by R. Griffiths et al. [“Con-cavity of magnetization of an Ising ferromagnet in a positive external field”, )] to the dimer model. More precisely, we provide a measure-preserving map between double random currents (obtained as the sum of two independent random currents) on a planar graph and dimers on an associated bipartite graph. We also define a nesting field for the double random current, which, under this map, corresponds to the height function of the dimer model. As applications, we provide an alternative derivation of some of the bozonization rules ob-tained recently by J. Dubédat [“Exact bosonization of the Ising model”, ], and show that the spontaneous magnetization of the Ising model on a planar biperiodic graph vanishes at criticality.

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