After much success in using the double random current representation in the study of the Ising model, Duminil-Copin posed the question in 2016 of determining the (percolative) phase transition of the single random current. By relating the single random current to the loop O(1) model, we prove polynomial lower bounds for path probabilities (and infinite expectation of cluster sizes) for both the single random current and loop O(1) model corresponding to any supercritical Ising model on the hypercubic lattice. Thereby partially resolving the posed question.
In this talk, I will gently introduce graphical representations of the Ising model and their relations through the uniform even subgraph. Afterward, we discuss new results whose surprising proof takes inspiration from the toric code in quantum theory.
Based on joint work with Ulrik Tinggaard Hansen and Boris Kjær: https://arxiv.org/abs/2306.05130.