Localization and Perron--Frobenius theory for directed polymers

TL;DR

This paper investigates directed polymers in a deterministic potential with random signs, establishing the existence of unique eigenfunctions and Gibbs measures, and demonstrating how local interactions determine macroscopic behavior.

Contribution

It introduces a novel analysis of directed polymers with a deterministic profile and random signs, proving uniqueness of eigenfunctions and Gibbs measures in this setting.

Findings

Existence of a unique positive cocycle eigenfunction.

Uniqueness of the infinite volume Gibbs measure.

Local interactions determine macroscopic structure.

Abstract

We consider directed polymers in a random potential given by a deterministic profile with a strong maximum at the origin taken with random sign at each integer time. We study two main objects based on paths in this random potential. First, we use the random potential and averaging over paths to define a parabolic model via a random Feynman--Kac evolution operator. We show that for the resulting cocycle, there is a unique positive cocycle eigenfunction serving as a forward and pullback attractor. Secondly, we use the potential to define a Gibbs specification on paths for any bounded time interval in the usual way and study the thermodynamic limit and existence and uniqueness of an infinite volume Gibbs measure. Both main results claim that the local structure of interaction leads to a unique macroscopic object for almost every realization of the random potential.