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 $\mathrm{O}$(1) model. In this paper, we first prove that the uniform even subgraph of $\mathbb{Z}^d$ percolates for $d \geq 2$ using its characterisation as the Haar measure on the group of even graphs. We then tighten the result by showing that the loop $\mathrm{O}$(1) model on $\mathbb{Z}^d$ percolates for $d \geq 2$ on some interval $(1-\varepsilon,1]$. Finally, our main theorem is that the loop $\mathrm{O}$(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.