Random Even Graphs

We study a random even subgraph of a finite graph G with a general edge-weight p∈(0,1). We demonstrate how it may be obtained from a certain random-cluster measure on G, and we propose a sampling algorithm based on coupling from the past. A random even subgraph of a planar lattice undergoes a phase transition at the parameter-value 12 pc, where pc is the critical point of the $q=2$ random-cluster model on the dual lattice. The properties of such a graph are discussed, and are related to Schramm–Löwner evolutions (SLE).