Full-path localization of directed polymers

TL;DR

This paper demonstrates that in directed polymer models at low temperatures, the overlap between independent samples occurs uniformly along the entire path, not just on average, by identifying key trajectories in Gaussian environments.

Contribution

It establishes the full-path localization of overlaps in directed polymers, revealing that the Gibbs measure concentrates on paths with persistent overlap along the entire length.

Findings

Overlap occurs along the entire polymer path.

Gibbs measure concentrates on a finite set of distinguished trajectories.

Results hold in all dimensions for Gaussian environments.

Abstract

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.