  We consider discrete (time and space) random walks confined to the quarter
plane, with jumps only in directions $(i,j)$ with $i+j \geq 0$ and small
negative jumps, i.e., $i,j \geq -1$. These walks are called singular, and were
recently intensively studied from a combinatorial point of view. In this paper,
we show how the compensation approach introduced in the 90ies by Adan, Wessels
and Zijm may be applied to compute positive harmonic functions with Dirichlet
boundary conditions. In particular, in case the random walks have a drift with
positive coordinates, we derive an explicit formula for the escape probability,
which is the probability to tend to infinity without reaching the boundary
axes. These formulas typically involve famous recurrent sequences, such as the
Fibonacci numbers. As a second step, we propose a probabilistic interpretation
of the previously constructed harmonic functions and prove that they allow to
compute all positive harmonic functions of these singular walks. To that
purpose, we derive the asymptotics of the Green functions in all directions of
the quarter plane and use Martin boundary theory.
