  For a reductive group $G$ over a non-Archimedean local field (e.g $GL_n(
\mathbb{Q}_p )$ ), Jacquet's Whittaker function is essentially proportional to
a character of an irreducible representation of the Langlands dual group
$G^\vee( \mathbb{C} )$ ( a Schur function if $G = GL_n( \mathbb{Q}_p )$). We
propose a probabilistic approach to this claim, known as the
Shintani-Casselman-Shalika formula, when the group $G$ has at least one
minuscule cocharacter in the coweight lattice.
  Our presentation goes along the following lines. Thanks to a minuscule random
walk $W^{(z)}$ on the coweight lattice and a related random walk on the Borel
subgroup, we establish a Poisson kernel formula for the non-Archimedean
Whittaker function. The expression and its ingredients are similar to the one
previously obtained by the author in the Archimedean case. A simple
manipulation reduces the problem to evaluating the probability of $W^{(z)}$
never exiting the Weyl chamber. Then, an implementation of the reflection
principle forces the appearance of the Weyl character formula and therefore
retrieves characters of $G^\vee\left( \mathbb{C} \right)$.
  The construction of the random walk on the Borel subgroup requires some care.
It is extracted from a spherical random walk whose increments have a
distribution that can be understood as elements from the spherical Hecke
algebra.
