The Supercritical Loop O(1) and Random Current models: Uniqueness and Mixing

Much recent rigorous study of the classical ferromagnetic Ising model has been powered by its graphical representations, such as the random current and loop O(1) model (high temperature expansion). In this paper, we prove uniqueness of Gibbs measures and exponential ratio weak mixing for the loop O(1) and random current models corresponding to the supercritical Ising model on the hypercubic lattice ${\mathbb{Z}}^d$ in any dimension $d\ge 2$. The main technical innovation is to establish unique crossing events for conditional random-cluster measures by a delicate exploration coupling of Pisztora’s coarse-graining method across scales. The results generalise to $q$-flow models and have natural applications for gradient measures of $\mathbb{Z}\mathrm{/}q\mathbb{Z}$-gauge theories.