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 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 obtained recently by Dubédat, and show that the spontaneous magnetization of the Ising model on a planar biperiodic graph vanishes at criticality.
Highlights
The goal of this paper is to present a new connection between the Ising model and dimers through double random currents, and to show some of its applications
The link between dimers and the Ising model has a long history that we will not describe in detail here
[21] Peierls used the so-called low-temperature expansion of the model to show the existence of an order-disorder phase transition in the Ising model on Z2
Summary
The goal of this paper is to present a new connection between the Ising model and dimers through double random currents, and to show some of its applications. The link between dimers and the Ising model has a long history that we will not describe in detail here (we refer the reader to the extensive literature for more information). The articles that we choose to mention in the introduction are the ones directly relevant to our new connection
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
