  We use k-Schur functions to get the minimal boundary of the k-bounded
partition poset. This permits to describe the central random walks on affine
Grassmannian elements of type A and yields a polynomial expression for their
drift. We also recover Rietsch's parametriza-tion of totally nonnegative
unitriangular Toeplitz matrices without using quantum cohomology of flag
varieties. All the homeomorphisms we define can moreover be made explicit by
using the combinatorics of k-Schur functions and elementary computations based
on Perron-Frobenius theorem.
