  The Wiener index of a finite graph G is the sum over all pairs (p, q) of
vertices of G of the distance between p and q. When P is a finite poset, we
define its Wiener index as the Wiener index of the graph of its Hasse diagram.
In this paper, we find exact expressions for the Wiener indices of the
distributive lattices of order ideals in minuscule posets. For infinite
families of such posets, we also provide results on the asymptotic distribution
of the distance between two random order ideals.
