  We study the spectrum and dynamics of a one-dimensional discrete Dirac
operator in a random potential obtained by damping an i.i.d. environment with
an envelope of type $n^{-\alpha}$ for $\alpha>0$. We recover all the spectral
regimes previously obtained for the analogue Anderson model in a random
decaying potential, namely: absolutely continuous spectrum in the
super-critical region $\alpha>\frac12$; a transition from pure point to
singular continuous spectrum in the critical region $\alpha=\frac12$; and pure
point spectrum in the sub-critical region $\alpha<\frac12$. From the dynamical
point of view, delocalization in the super-critical region follows from the
RAGE theorem. In the critical region, we exhibit a simple argument based on
lower bounds on eigenfunctions showing that no dynamical localization can occur
even in the presence of point spectrum. Finally, we show dynamical localization
in the sub-critical region by means of the fractional moments method and
provide control on the eigenfunctions.
