  The quantum and classical behaviors of two-dimensional (2D) alternative
quantum walk (AQW) in the presence of decoherence have been discussed in
detail. For any kinds of decoherence, the analytic expressions for the moments
of position distribution of AQW have been obtained. Taking the broken line
noise and coin-decoherence as examples of decoherence, we find that when the
decoherence only emerges in one direction, the anisotropic position
distribution pattern appears, and not all the motions of quantum walkers
exhibit the transition from quantum to classical behaviors. The correlations
between the walkers and the coin in 2D AQW have been discussed. The anisotropic
correlations between walkers and coin have been revealed in the presence of
decoherence.
