Universal principles of moiré band structures

Moir\'e materials provide a highly tunable environment for the realization of band structures with engineered physical properties. Specifically, moir\'e Fermi surface flat bands - synthetic correlated phases have unit cells containing thousands atoms and tantalizingly complex structures. In this paper we show that statistical principles go long way in explaining universal properties these systems. Our approach builds on three conceptual elements: presence quantum chaos caused by effective irregularity atomic configurations short length scales, Anderson localization momentum space, approximate crystalline symmetries. Which dominates depends material parameters such as extension or strength lattice potential. The phenomenological consequences competition are predictions characteristic group velocity bands, primary indicator their average flatness. addition to generic features, identify outside context, notably almost close extrema unperturbed spectra, celebrated zero energy `magic angle' where latter require exceptionally fine tuned parameters.