Joint localization of directed polymers

TL;DR

This paper establishes conditions under which multiple directed polymers in a random environment localize jointly in a common region, extending understanding of localization phenomena in polymer models.

Contribution

It provides new sufficient conditions for joint localization in directed polymers, including models with simple symmetric and Gaussian random walks, and proves strong disorder in continuous models.

Findings

Joint localization occurs for polymers with symmetric or Gaussian reference walks.

Strong disorder property holds for a broad class of continuous polymer models.

Single polymer localization is implied by strong disorder in these models.

Abstract

We consider $(1+1)$-dimensional directed polymers in a random potential and provide sufficient conditions guaranteeing joint localization. Joint localization means that for typical realizations of the environment, and for polymers started at different starting points, all the associated endpoint distributions localize in a common random region that does not grow with the length of the polymer. In particular, we prove that joint localization holds when the reference random walk of the polymer model is either a simple symmetric lattice walk or a Gaussian random walk. We also prove that the very strong disorder property holds for a large class of space-continuous polymer models, implying the usual single polymer localization.