  We introduce a family of two-dimensional reflected random walks in the
positive quadrant and study their Martin boundary. While the minimal boundary
is systematically equal to a union of two points, the full Martin boundary
exhibits an instability phenomenon, in the following sense: if some parameter
associated to the model is rational (resp.\ non-rational), then the Martin
boundary is countable, homeomorphic to $\mathbb Z\cup\{\pm\infty\}$ (resp.\
uncountable, homeomorphic to $\mathbb R\cup\{\pm\infty\}$). Such instability
phenomena are very rare in the literature. Along the way of proving this
result, we obtain several precise estimates for the Green functions of
reflected random walks with escape probabilities along the boundary axes and an
arbitrarily large number of inhomogeneity domains. Our methods mix
probabilistic techniques and an analytic approach for random walks with large
jumps in dimension two.
