Andreas Schäfer

We study random Quantum Walks (QWs) on general lattices. Every vertex is equipped with a unitary scattering matrix, which governs the local dynamics. We associate to each vertex an i.i.d. random phase and define the random QW by multiplying every scattering matrix with the corresponding random phase. We obtain dynamical localization under the condition that the scattering matrices are sufficiently close to the identity, which induces full localization. Under dynamical localization we understand the property that the probability to move from a lattice site x to another site y decreases on average exponentially in the distance |x-y|, independently of how many steps the Quantum Walker may take. One commonly derives dynamical localization from exponential decay of the fractional moments of the resolvent using a rank one analysis. However, this framework requires a high amount of randomness (deg(x) random variables per vertex x) and is not suitable for our model. We present a novel approach to link fractional moment estimates to dynamical localization, using eigenfunction correlators.