We consider families of random non-unitary contraction operators defined as
deformations of CMV matrices which appear naturally in the study of random
quantum walks on trees or lattices. We establish several deterministic and
almost sure results about the location and nature of the spectrum of such
non-normal operators as a function of their parameters. We relate these results
to the analysis of certain random quantum walks, the dynamics of which can be
studied by means of iterates of such random non-unitary contraction operators.Comment: updated version, to appear in Annales Henri Poincar