Extrema of 3D Potts interfaces

The interface between the plus and minus phases in the low temperature 3D Ising model has been intensely studied since Dobrushin's pioneering works in the early 1970's established its rigidity. Advances in the last decade yielded the tightness of the maximum of the interface of this Ising model on the cylinder of side length $n$, around a mean that is asymptotically $c\log n$ for an explicit $c$ (temperature dependent). In this work, we establish analogous results for the 3D Potts and random cluster (FK) models. Compared to 3D Ising, the Potts model and its lack of monotonicity form obstacles for existing methods, calling for new proof ideas, while its interfaces (and associated extrema) exhibit richer behavior. We show that the maxima and minima of the interface bounding the blue component in the 3D Potts interface, and those of the interface bounding the bottom component in the 3D FK model, are governed by 4 different large deviation rates, whence the corresponding global extrema feature 4 distinct constants $c$ as above. Due to the above obstacles, our methods are initially only applicable to 1 of these 4 interface extrema, and additional ideas are needed to recover the other 3 rates given the behavior of the first one.