The model is a classical model of ferromagnetism in statistical mechanics that has its origins in Euclidean quantum field theory. It is amongst the simplest models of unbounded spin that is expected to be in the Ising universality class. In these two talks, we will describe recent progress that we made with Christoforos Panagiotis (Université de Genève) and Franco Severo (ETH Zürich) towards understanding the critical behaviour of in dimensions .
In the first part, we will discuss a new geometric representation for called random tangled currents. This is the natural extension of the celebrated random current representation for the Ising model that has been the basis of many beautiful results concerning its critical behaviour since the seminal work of Aizenman in the ‘80s. At the heart of these results is the switching lemma. We prove that random tangled currents satisfy an analogous switching principle. In the second part, we will discuss the proof of continuity of the phase transition for in using random tangled currents. In particular, we will show how to adapt percolation-based techniques to this geometric representation. In our setting, significant conceptual and technical difficulties arise from the unboundedness of spin and lack of exact combinatorial symmetries in the model.