  We consider a family of random walks killed at the boundary of the Weyl
chamber of the dual of $\rm{Sp}(4)$, which in addition satisfies the following
property: for any $n\geq 3$, there is in this family a walk associated with a
reflection group of order $2n$. Moreover, the case $n=4$ corresponds to a
process which appears naturally by studying quantum random walks on the dual of
$\rm{Sp}(4)$. For all the processes belonging to this family, we find the exact
asymptotic of the Green functions along all infinite paths of states as well as
that of the absorption probabilities along the boundaries.
