Spectral Transition for Random Quantum Walks on Trees

We consider random quantum walks on a homogeneous tree of degree $3$ describing the discrete time evolution of a quantum particle with internal degree of freedom in $\mathbb C^3$ hopping on the neighbouring sites of the tree in presence of static disorder. The one time step random unitary evolution operator of the particle depends on a unitary matrix $C$ in $U(3)$ which monitors the strength of the disorder. We show the existence of open sets of matrices in $U(3)$ for which the random evolution has either pure point spectrum almost surely or purely absolutely continuous spectrum. We also establish properties of the spectral diagram which provide a description of the spectral transition driven by $C$ in $U(3)$.