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Spectral Transition for Random Quantum Walks on Trees

We define and analyze random quantum walks on homogeneous trees of degree $q\geq 3$. Such walks describe the discrete time evolution of a quantum particle with internal degree of freedom in $\C^q$ hopping on the neighboring 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 $C\in U(q)$ which monitors the strength of the disorder. We prove for any $q$ that there exist open sets of matrices in $U(q)$ for which the random evolution has either pure point spectrum almost surely or purely absolutely continuous spectrum, thereby showing the existence of a spectral transition driven by $C\in U(q)$. For $q\in\{3,4\}$, we establish properties of the spectral diagram which provide a description of the spectral transition.