  Quantum walks on graphs can model physical processes and serve as efficient
tools in quantum information theory. Once we admit random variations in the
connectivity of the underlying graph, we arrive at the problem of percolation,
where the long-time behaviour appears untreatable with direct numerical
methods. We develop novel analytic methods based on the theory of random
unitary operations which help us to determine explicitly the asymptotic
dynamics of quantum walks on 2D finite integer lattices with percolation. Based
on this theory we find new unexpected features of percolated walks like
asymptotic position inhomogeneity or special directional symmetry breaking.
