  A random walk in $Z_+^2$ spatially homogeneous in the interior, absorbed at
the axes, starting from an arbitrary point $(i_0,j_0)$ and with step
probabilities drawn on Figure 1 is considered. The trivariate generating
function of probabilities that the random walk hits a given point $(i,j)\in
Z_+^2$ at a given time $k\geq 0$ is made explicit. Probabilities of absorption
at a given time $k$ and at a given axis are found, and their precise asymptotic
is derived as the time $k\to\infty$. The equivalence of two typical ways of
conditioning this random walk to never reach the axes is established. The
results are also applied to the analysis of the voter model with two candidates
and initially, in the population $Z$, four connected blocks of same opinions.
Then, a citizen changes his mind at a rate proportional to the number of its
neighbors that disagree with him. Namely, the passage from four to two blocks
of opinions is studied.
