On the double random current nesting field

Abstract

We relate the planar random current representation introduced by Griffiths, Hurst and Sherman 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 construct 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 obtained recently by Dub\'edat, and show that the spontaneous magnetization of the Ising model on a planar biperiodic graph vanishes at criticality.