  We study the asymptotic position distribution of general quantum walks on a
lattice, including walks with a random coin, which is chosen from step to step
by a general Markov chain. In the unitary (i.e., non-random) case, we allow any
unitary operator, which commutes with translations, and couples only sites at a
finite distance from each other. For example, a single step of the walk could
be composed of any finite succession of different shift and coin operations in
the usual sense, with any lattice dimension and coin dimension. We find
ballistic scaling, and establish a direct method for computing the asymptotic
distribution of position divided by time, namely as the distribution of the
discrete time analog of the group velocity. In the random case, we let a Markov
chain (control process) pick in each step one of finitely many unitary walks,
in the sense described above. In ballistic order we find a non-random drift,
which depends only on the mean of the control process and not on the initial
state. In diffusive scaling the limiting distribution is asymptotically
Gaussian, with a covariance matrix (diffusion matrix) depending on momentum.
The diffusion matrix depends not only on the mean but also on the transition
rates of the control process. In the non-random limit, i.e., when the coins
chosen are all very close, or the transition rates of the control process are
small, leading to long intervals of ballistic evolution, the diffusion matrix
diverges. Our method is based on spatial Fourier transforms, and the first and
second order perturbation theory of the eigenvalue 1 of the transition operator
for each value of the momentum.
