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The interface between the plus and minus phases in the low temperature 3DIsing model has been intensely studied since Dobrushin's pioneering works inthe early 1970's established its rigidity. Advances in the last decade yieldedthe tightness of the maximum of the interface of this Ising model on thecylinder of side length $n$, around a mean that is asymptotically $c\log n$ foran explicit $c$ (temperature dependent). In this work, we establish analogousresults for the 3D Potts and random cluster (FK) models. Compared to 3D Ising,the Potts model and its lack of monotonicity form obstacles for existingmethods, calling for new proof ideas, while its interfaces (and associatedextrema) exhibit richer behavior. We show that the maxima and minima of theinterface bounding the blue component in the 3D Potts interface, and those ofthe interface bounding the bottom component in the 3D FK model, are governed by4 different large deviation rates, whence the corresponding global extremafeature 4 distinct constants $c$ as above. Due to the above obstacles, ourmethods are initially only applicable to 1 of these 4 interface extrema, andadditional ideas are needed to recover the other 3 rates given the behavior ofthe first one.

Extrema of 3D Potts interfaces