Random even graphs and the Ising model

We explore the relationship between the Ising model with inverse temperature β, the q = 2 random-cluster model with edge-parameter p = 1− e−2β , and the random even subgraph with edge-parameter 12p. For a planar graph G, the boundary edges of the + clusters of the Ising model on the planar dual of G forms a random even subgraph of G. A coupling of the random even subgraph of G and the q = 2 random-cluster model on G is presented, thus extending the above observation to general graphs. A random even subgraph of a planar lattice undergoes a phase transition at the parameter-value 12pc, where pc is the critical point of the q = 2 random-cluster model on the dual lattice. These results are motivated in part by an exploration of the socalled random-current method utilised by Aizenman, Barsky, Fernández and others to solve the Ising model on the d-dimensional hypercubic lattice.