  The first main goal of this article is to give a new metrization of the
Mukherjee--Varadhan topology, recently introduced as a translation-invariant
compactification of the space of probability measures on Euclidean spaces. This
new metrization allows us to achieve our second goal which is to extend the
recent program of Bates and Chatterjee on localization for the endpoint
distribution of discrete directed polymers to polymers based on general random
walks in Euclidean spaces. Following their strategy, we study the asymptotic
behavior of the endpoint distribution update map and study the set of its
distributional fixed points satisfying a variational principle. We show that
the distributdion concentrated on the zero measure is a unique element in this
set if and only if the system is in the high temperature regime. This enables
us to prove that the asymptotic clustering (a natural continuous analogue of
the asymptotic pure atomicity property) holds in the low temperature regime and
that the endpoint distribution is geometrically localized with positive density
if and only if the system is in the low temperature regime.
