Ergodic properties of one-dimensional incommensurate bilayer materials

We consider one-dimensional deterministic and random tight-binding Hamiltonians modeling electronic properties of twisted bilayer materials. When the twisted structure is incommensurate, we prove convergence of the density of states measure in the thermodynamic limit and Pastur’s theorem on shift-invariance of the spectrum. Our results extend those of Massatt et al. and Cancès et al. in allowing for randomness. We provide numerical density of states computations for the operators we consider.