Ballistic transport in periodic and random media

Abstract: We prove ballistic transport of all orders, that is, $\lVert x^m\mathrm{e}^{-\mathrm{i}tH}\psi\rVert\asymp t^m$, for the following models: the adjacency matrix on $\mathbb{Z}^d$, the Laplace operator on $\mathbb{R}^d$, periodic Schr\"odinger operators on $\mathbb{R}^d$, and discrete periodic Schr\"odinger operators on periodic graphs. In all cases we give the exact expression of the limit of $\lVert x^m\mathrm{e}^{-\mathrm{i}tH}\psi\rVert/t^m$ as $t\to+\infty$. We then move to universal covers of finite graphs (these are infinite trees) and prove ballistic transport in mean when the potential is lifted naturally, giving a periodic model, and when the tree is endowed with random i.i.d.\ potential, giving an Anderson model. The limiting distributions are then discussed, enriching the transport theory. Some general upper bounds are detailed in the appendix.