On the double random current nesting field

Hugo Duminil-Copin, Marcin Lis
2019 Probability theory and related fields  
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 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
more » ... derivation of some of the bozonization rules obtained recently by Dubédat, and show that the spontaneous magnetization of the Ising model on a planar biperiodic graph vanishes at criticality.
doi:10.1007/s00440-019-00899-0 fatcat:gpvxvs4i2naofnohv2fnxi3eey