Proof of the path localization conjecture for directed polymers

TL;DR

This paper proves the long-standing conjecture that directed polymers in random environments localize around a favorite path at low temperatures, advancing understanding of their behavior.

Contribution

It provides a rigorous proof of the path localization conjecture for directed polymers, a major open problem in the field.

Findings

Confirmed localization of directed polymers at low temperature.

Established a precise mathematical formulation of the conjecture.

Provided the first proof of this conjecture in the literature.

Abstract

It is a well-known open problem in the literature on random polymers to show that a directed polymer in random environment localizes around a favorite path at low temperature. A precise statement of this conjecture is formulated and proved in this article.