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Ballistic Behavior for Random Schrödinger Operators on the Bethe Strip

The Bethe Strip of width $m$ is the cartesian product $\B\times\{1,...,m\}$, where $\B$ is the Bethe lattice (Cayley tree). We consider Anderson-like Hamiltonians $H_λ=\frac12 Δ\otimes 1 + 1 \otimes A+λ\Vv$ on a Bethe strip with connectivity $K \geq 2$, where $A$ is an $m\times m$ symmetric matrix, $\Vv$ is a random matrix potential, and $λ$ is the disorder parameter. Under certain conditions on $A$ and $K$, for which we previously proved the existence of absolutely continuous spectrum for small $λ$, we now obtain ballistic behavior for the spreading of wave packets evolving under $H_λ$ for small $λ$.