Analysis and Mathematical Physics

This talk will be about a ferromagnetic spin system called the Blume-Capel model. It was introduced in the '60s to model an exotic multi-critical phase transition observed in the magnetisation of uranium oxide. Mathematically speaking, the model can be viewed as an Ising model coupled to a site percolation (with annealed disorder), thus combining two of the most beautiful models in statistical physics. In spite of this, little was rigorously known about this model's critical behaviour. I will explain how modern stochastic geometric methods, based on percolation representations of spin models, seem to be useful to analyse the critical behaviour of the Blume-Capel model. In particular, I will explain a result obtained in recent joint work with Dmitry Krachun and Christoforos Panagiotis (both at Geneva) that establishes the existence of a so-called tricritical point in any dimension $d\geq 3$. My ultimate goal, however, is to convince everyone that this is just the beginning of the story!