Andreas Schaefer

We study Quantum Walks (QW) on the hexagonal lattice. The random unitary operator describing the QW is given by the composition of a shift and coin operator, together with a random phase that depends on the lattice site the Walker is in. We will prove dynamical localization under the condition that the coin matrix used to define the coin operator is close enough to the permutation matrices that correspond to the permutations σ = (1 2 3) or σ = (1 3 2) and induce 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 Walker may take. Dynamical localization leads to the operator having pure point spectrum, which, using the RAGE Theorem, implies that the QW will stay in some bounded area with high probability. We will prove dynamical localization by showing exponential decay of the fractional moments of the resolvent.