UVictoria Probability and Dynamics Seminar: Frederik R. Klausen

Much of the recent rigorous progress on the classical Ising model was driven by new detailed understanding of its stochastic geometric representations. 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.

Using that the loop O(1) model is the uniform even graph of the random cluster 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. The method partly extends to all positive integers q, where the analogue of the loop O(1) model is the q-flow model.

In this talk, I will introduce graphical representations of the Potts and Ising model and their many couplings followed by a discussion of new results whose surprising proof takes inspiration from the toric code in quantum theory.

Based on: https://link.springer.com/article/10.1007/s00220-025-05297-3 and https://arxiv.org/abs/2506.10765