The Uniform Even Subgraph and Its Connection to Phase Transitions of Graphical Representations of the Ising Model

The uniform even subgraph is intimately related to the Ising model, the random-cluster model, the random current model, and the loop \(\textrm{O}\)(1) model. In this paper, we first prove that the uniform even subgraph of \(\mathbb {Z}^d\) percolates for \(d \ge 2\) using its characterisation as the Haar measure on the group of even graphs. We then tighten the result by showing that the loop \(\textrm{O}\)(1) model on \(\mathbb {Z}^d\) percolates for \(d \ge 2\) for edge-weights x lying in some interval \((1-\varepsilon ,1]\). Finally, our main theorem is that the loop \(\textrm{O}\)(1) model and random current models corresponding to a supercritical Ising model are always at least critical, in the sense that their two-point correlation functions decay at most polynomially and the expected cluster sizes are infinite.