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              <p>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.</p>
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              <p>Nous introduisons une famille de marches al\'eatoires en dimension deux, r\'efl\'echies au bord du quart de plan positif, et \'etudions leur fronti\`ere de Martin. Tandis que leur fronti\`ere minimale est syst\'ematiquement une union de deux points, nous montrons que la fronti\`ere de Martin compl\`ete est intrins\`equement instable, au sens suivant : lorsqu'un certain param\`etre associ\'e au mod\`ele s'av\`ere rationnel (respectivement non rationnel), la fronti\`ere de Martin est alors d\'enombrable et hom\'eomorphe \`a $\mathbb Z\cup\{\pm\infty\}$ (respectivement non d\'enombrable et hom\'eomorphe \`a $\mathbb R\cup\{\pm\infty\}$). De tels ph\'enom\`enes d'instabilit\'e sont rares dans la litt\'erature. Les d\'emonstrations contiennent plusieurs estim\'ees pr\'ecises pour des fonctions de Green de marches al\'eatoires r\'efl\'echies avec probabilit\'e de fuite le long des axes, poss\'edant en outre un nombre infini de domaines d'inhomog\'en\'eit\'e. Nos m\'ethodes m\'elangent des techniques probabilistes avec une approche analytique pour des marches al\'eatoires avec grands pas en dimension deux.</p>
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