Under a natural asumption on the drift, the law of the simple random walk on the multidimensional first quadrant conditioned to always stay in the first octant was obtained by O'Connell in [O]. It coincides with that of the image of the simple random walk under the multidimensional Pitman transform and can be expressed in terms of specializations of Schur functions. This result has been generalized in [LLP1] and [LLP2] for a large class of random walks on weight lattices defined from representations of Kac-Moody algebras and their conditionings to always stay in Weyl chambers. In these various works, the drift of the considered random walk is always assumed in the interior of the cone. In this paper, we introduce for some zero drift random walks defined from minuscule representations a relevant notion of conditioning to stay in Weyl chambers and we compute their laws. Namely, we consider the conditioning for these walks to stay in these cones until an instant we let tend to infinity. We also prove that the laws so obtained can be recovered by letting the drift tend to zero in the transitions matrices obtained in [LLP1]. We also conjecture our results remain true in the more general case of a drift in the frontier of the Weyl chamber.
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