On the double random current nesting field

TL;DR

This paper establishes a connection between double random current models and dimer models on planar graphs, introducing a nesting field that links to the dimer height function, with applications to Ising model criticality.

Contribution

It provides a measure-preserving map between double random currents and dimers, and constructs a nesting field linking these models, offering new insights into their structure and phase transitions.

Findings

Derived an alternative bozonization rule.

Showed spontaneous magnetization vanishes at criticality.

Established a measure-preserving correspondence between models.

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.