Quantum ergodicity for periodic graphs

We prove quantum ergodicity for a family of periodic Schrödinger operators $H$ on periodic graphs. This means that most eigenfunctions of $H$ on large finite periodic graphs are equidistributed in some sense, hence delocalized. Our results cover the adjacency matrix on ${\mathbb{Z}}^d$, the triangular lattice, the honeycomb lattice, Cartesian products and periodic Schrödinger operators on ${\mathbb{Z}}^d$. The theorem applies more generally to any periodic Schrödinger operator satisfying an assumption on the Floquet eigenvalues.