QLunch: Ulrik Thingaard Hansen

We introduce the Ising and Potts models and how to study them using the Fortuin–Kasteleyn (FK) representation through the Edwards-Sokal coupling. 

We introduce critical phenomena with some examples from the two-dimensional case and describe how the FK-representation can be used to study the model. 

Furthermore, the representation adapts to the setting where the models are exposed to an external field of strength h > 0. So in this model, which is also known as the random-cluster model, the Kertész line separates the two regions of parameter space according to the existence of an infinite cluster in Z^d. 

This signifies a geometric phase transition between the ordered and disordered phases even in cases where a thermodynamic phase transition does not occur. Thus, the graphical representation lets us extend the notion of phase transition. 

We explain recent results on strict monotonicity and continuity of the Kertész line. Furthermore, investigate rigorous bounds that are asymptotically correct in the limit h → 0 complementing the older bounds, which were asymptotically correct for h → ∞. 

This is based on https://arxiv.org/abs/2206.07033 and is joint work with Frederik Ravn Klausen. "