  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.
