| Abstract: |
| In this talk, we present some results on ergodicity for homogeneous discrete-time quantum walks on the integer lattice. The most significant result holds in dimension one and gives a complete equivalence between the absolutely continuous spectrum of the unitary operator encoding the walk and the equidistribution of its dynamics in position space. In higher dimensions, we give a criterion for full and partial ergodicity in terms of a finer property of the spectrum which we dub `No Repeating Graphs`, and distinguish how strongly the equidistribution is taking place (weak convergence vs total variation). We also present applications of our results to the ergodicity of eigenvectors for Schr\{o}dinger operators on $\mathbb{Z}$-periodic graphs. |
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