Weak coupling limits for directed polymers in tube environments

TL;DR

This paper investigates how directed polymers behave in a growing tube environment, revealing phase transitions between disorder regimes as the environment's width and temperature change.

Contribution

It introduces a model interpolating between pinning and directed polymers, establishing weak coupling limits across all dimensions with varying tube widths.

Findings

Identifies regimes of disorder irrelevance, marginal relevance, and relevance.

Establishes weak coupling limits for the partition function.

Shows phase transitions depending on tube width and temperature rate.

Abstract

In this paper, we study a model of directed polymers in random environment, where the environment is restricted to a time-space tube whose spatial width grows polynomially with time. It can be viewed as an interpolation between the disordered pinning model and the classic directed polymer model. We prove weak coupling limits for the directed polymer partition functions in such tube environments in all dimensions, as the inverse temperature vanishes at a suitable rate. As the tube width varies, transitions between regimes of disorder irrelevance, marginal relevance and disorder relevance are observed.