  Nearest neighbor random walks in the quarter plane that are absorbed when
reaching the boundary are studied. The cases of positive and zero drift are
considered. Absorption probabilities at a given time and at a given site are
made explicit. The following asymptotics for these random walks starting from a
given point $(n_0,m_0)$ are computed : that of probabilities of being absorbed
at a given site $(i,0)$ [resp. $(0,j)$] as $i\to \infty$ [resp. $j \to
\infty$], that of the distribution's tail of absorption time at x-axis [resp.
y-axis], that of the Green functions at site $(i,j)$ when $i,j\to \infty$ and
$j/i \to \tan \gamma$ for $\gamma \in [0, \pi/2]$. These results give the
Martin boundary of the process and in particular the suitable Doob
$h$-transform in order to condition the process never to reach the boundary.
They also show that this $h$-transformed process is equal in distribution to
the limit as $n\to \infty$ of the process conditioned by not being absorbed at
time $n$. The main tool used here is complex analysis.
