Non-uniqueness of phase transitions for graphical representations of the Ising model on tree-like graphs

We consider the graphical representations of the Ising model on tree-like graphs. We construct a class of graphs on which the loop O(1) model and the single random current exhibit a non-unique phase transition with respect to the inverse temperature, highlighting the non-monotonicity of both models. It follows from the construction that there exist infinite graphs G ⊆ G′ such that the uniform even subgraph of G′ percolates and the uniform even subgraph of G does not. We also show that on the wired d-regular tree, the phase transitions of the loop O(1), the single random current, and the random-cluster models are all unique and coincide.