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 , where 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.
This research was supported by NSF grant DMS-1902734.
I am indebted to Sourav Chatterjee for suggesting the consideration of a time-dependent inverse temperature. This idea was the inspiration leading to the present article. I also thank Francis Comets for valuable discussion during the workshop on “Self-interacting random walks, Polymers and Folding” held at Centre International de Rencontres Mathématiques, and Hubert Lacoin for directing my attention to [10, Sec. 7]. Finally, I am very grateful to the referees for their corrections and suggestions, which improved the exposition.
"Full-path localization of directed polymers." Electron. J. Probab. 26 1 - 24, 2021. https://doi.org/10.1214/21-EJP641