# Phase transitions of graphical representations of the Ising model

# Phase transitions of graphical representations of the Ising model

Much of the recent rigours progress on the classical Ising model was driven by new detailed understanding of its stochastic geometric representations - in particular the random current representation. Motivated by the problem of establishing exponential decay of truncated correlations of the supercritical Ising model in any dimension,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 of 0) for both the single random current and loop O(1) model corresponding to any supercritical Ising model on the hypercubic lattice.

In this talk, I will gently introduce graphical representations of the Ising model and motivate the theory of percolation followed by a discussion of 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