Full-path localization of directed polymers

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