Ballistic Behavior for Random Schr\"odinger Operators on the Bethe Strip

The Bethe Strip of width $m$ is the cartesian product $\text{B}\times \{1,\mathrm{.}\mathrm{.}\mathrm{.},m\}$, where $\text{B}$ is the Bethe lattice (Cayley tree). We consider Anderson-like Hamiltonians $H_{\lambda }=\frac{1}{2}\mathrm{\Delta }\otimes 1+1\otimes A+\lambda \text{Vv}$ on a Bethe strip with connectivity $K\ge 2$, where $A$ is an $m\times m$ symmetric matrix, $\text{Vv}$ is a random matrix potential, and $\lambda$ is the disorder parameter. Under certain conditions on $A$ and $K$, for which we previously proved the existence of absolutely continuous spectrum for small $\lambda$, we now obtain ballistic behavior for the spreading of wave packets evolving under $H_{\lambda }$ for small $\lambda$.