  Certain polymer models are known to exhibit path localization in the sense
that at low temperatures, the average fractional overlap of two independent
samples from the Gibbs measure is bounded away from $0$. Nevertheless, the
question of where along the path this overlap takes place has remained
unaddressed. In this article, we prove that on linear scales, overlap occurs
along the entire length of the polymer. Namely, we consider time intervals of
length $\varepsilon N$, where $\varepsilon>0$ is fixed but arbitrarily small.
We then identify a constant number of distinguished trajectories such that the
Gibbs measure is concentrated on paths having, with one of these distinguished
paths, a fixed positive overlap simultaneously in every such interval. This
result is obtained in all dimensions for a Gaussian random environment by using
a recent non-local result as a key input.
