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

Abstract

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.