  The limit weak order on an affine Weyl group was introduced by Lam and
Pylyavskyy in their study of total positivity for loop groups. They showed that
in the case of the affine symmetric group the minimal elements of this poset
coincide with the infinite fully commutative reduced words and with infinite
powers of Coxeter elements. We answer several open problems raised there by
classifying minimal elements in all affine types and relating these elements to
the classes of fully commutative and Coxeter elements. Interestingly, the
infinite fully commutative elements correspond to the minuscule and cominuscule
nodes of the Dynkin diagram, while the infinite Coxeter elements correspond to
a single node, which we call the heavy node, in all affine types other than
type $A$.
